Stochastic Methods for Boundary Value Problems: Numerics for High-dimensional PDEs and Applications 9783110479454, 9783110479065

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Karl K. Sabelfeld and Nikolai A. Simonov Stochastic Methods for Boundary Value Problems

Also of Interest Random Fields and Stochastic Lagrangian Models: Analysis and Applications in Turbulence and Porous Media Karl K. Sabelfeld, 2012 ISBN 978-3-11-029664-8, e-ISBN 978-3-11-029681-5

Spherical and Plane Integral Operators for PDEs: Construction, Analysis, and Applications Karl K. Sabelfeld, Irina A. Shalimova, 2013 ISBN 978-3-11-031529-5, e-ISBN 978-3-11-031533-2

Monte Carlo Methods and Applications Karl K. Sabelfeld (Managing Editor) ISSN 0929-9629, e-ISSN 1569-3961

Computational Methods in Applied Mathematics Carsten Carstensen (Editor-in-Chief) ISSN 1609-4840, e-ISSN 1609-9389

Radon Series on Computational and Applied Mathematics Ulrich Langer, Hansjörg Albrecher, Heinz W. Engl, Ronald H. W. Hoppe, Karl Kunisch, Harald Niederreiter, Christian Schmeiser (Ed.) ISSN 1865-3707

Karl K. Sabelfeld and Nikolai A. Simonov

Stochastic Methods for Boundary Value Problems | Numerics for High-Dimensional PDEs and Applications

Authors Prof. Dr. Karl K. Sabelfeld Novosibirsk University (NSU) Russian Academy of Sciences Siberian Branch Institute of Computational Mathematics and Mathematical Geophysics Lavrentjeva 6 630090 Novosibirsk RUSSIA [email protected] Nikolai A. Simonov Russian Academy of Sciences Siberian Branch Institute of Computational Mathematics and Mathematical Geophysics Lavrentjeva 6 630090 Novosibirsk RUSSIA [email protected]

ISBN 978-3-11-047906-5 e-ISBN (PDF) 978-3-11-047945-4 e-ISBN (EPUB) 978-3-11-047916-4 Set-ISBN 978-3-11-047946-1 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2016 Walter de Gruyter GmbH, Berlin/Boston Cover image: Rui Tho/Hemera/thinkstock Typesetting: Konvertus, Haarlem Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface The monograph is devoted to random walk-based stochastic algorithms for solving high-dimensional boundary value problems of mathematical physics and chemistry. The book includes stochastic algorithms of two classes of random walk methods: (1) methods where the random walks live on the boundary and (2) methods where the random walks are Markov chains of random points inside the domain. The random walk-on-boundary methods are presented in Chapters 1–6 for the main boundary value problems of the heat, electrostatics and elastostatics potential theory. The methods based on random walks inside the domain are intrinsically related with the probabilistic representations of solutions to PDEs in the form of Wiener path integrals. Therefore, different ideas from probabilistic representations, such as the first passage time, exit point and survival probability concepts, are used to construct highly efficient stochastic algorithms. We show how these methods can be used to solve some applied problems, in particular the simulation of kinetics of electron-hole recombination in semiconductors, cathodoluminescence imaging method, capacitance calculations, evaluation of electrostatic energy for macromolecules in salt solutions and others. The stochastic algorithms based on random walks are grid free; they are all scalable and can be easily parallelized. Along with the probabilistic representations, we use also a functional approach based on Green functions and local integral equations, which are equivalent to the original boundary value problems. The book is written for mathematicians who work in the field of partial differential and integral equations, for physicists and engineers dealing with computational methods and applied probability, and for students and postgraduates studying mathematical physics and numerical mathematics. The support of Russian Science Foundation under grant No. 14-11-00083 is kindly acknowledged.

Contents 1

Introduction | 1

2 2.1 2.2 2.3

Random walk algorithms for solving integral equations | 8 Conventional Monte Carlo scheme | 8 Biased estimators | 14 Linear-fractional transformations and their relations to iterative processes | 16 Asymptotically unbiased estimators based on singular approximations | 23 Integral equation of the first kind | 30

2.4 2.5 3 3.1

3.6 3.7 3.7.1 3.8 3.9

Random walk-on-boundary algorithms for the Laplace equation | 35 Newton potentials and boundary integral equations of the electrostatics | 35 The interior Dirichlet problem and isotropic random walk-on-boundary process | 37 Solution of the Neumann problem | 43 Random estimators for the exterior Dirichlet problem | 50 Third BVP and alternative methods of solving the Dirichlet problem | 55 Inhomogeneous problems | 60 Continuity BVP | 62 Walk on boundary for the continuity problem | 64 Calculation of the solution derivatives near the boundary | 65 Normal derivative of a double-layer potential | 69

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Walk-on-boundary algorithms for the heat equation | 72 Heat potentials and Volterra boundary integral equations | 72 Nonstationary walk-on-boundary process | 74 The Dirichlet problem | 77 The Neumann problem | 80 Third BVP | 81 Unbiasedness and variance of the walk-on-boundary algorithms | 85 The cost of the walk-on-boundary algorithms | 90 Inhomogeneous heat equation | 92 Calculation of derivatives on the boundary | 94

3.2 3.3 3.4 3.5

viii | Contents

5 5.1 5.2 5.3

6 6.1 6.1.1 6.1.2 6.2 6.3 6.4 6.5 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.6 8 8.1 8.2 8.3 8.3.1 8.3.2

Spatial problems of elasticity | 99 Elastopotentials and systems of boundary integral equations of the elasticity theory | 99 First BVP and estimators for singular integrals | 102 Other BVPs for the Lamé equations and regular integral equations | 107 Variants of the random walk on boundary for solving stationary potential problems | 110 The Robin problem and the ergodic theorem | 110 Monte Carlo estimator for computing capacitance | 113 Computing charge density | 114 Stationary diffusion equation with absorption | 115 Multiply connected domains | 116 Stabilization method | 126 Nonlinear Poisson equation | 128 Splitting and survival probabilities in random walk methods and applications | 131 Introduction | 131 Survival probability for a sphere and an interval | 132 The reciprocity theorem for particle collection in the general case of Robin boundary conditions | 135 Splitting and survival probabilities | 138 Splitting probabilities for a finite interval and nanowire growth simulation | 138 Survival probability for a disc and the exterior of circular cylinder | 140 Splitting probabilities for concentric spheres and annulus | 141 Cathodoluminescence | 145 The random WOS and hemispheres algorithm | 147 Conclusion and discussion | 152 A random WOS-based KMC method for electron–hole recombinations | 153 Introduction | 153 The mean field equations | 155 Monte Carlo Algorithms | 157 Random WOS for the diffusion simulation | 157 Radiative and nonradiative recombination in the absence of diffusion | 160

Contents |

8.3.3 8.4 8.5 9 9.1 9.1.1 9.1.2 9.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 9.4.2 9.4.3 9.4.4 9.5 9.5.1

General case of radiative and nonradiative recombination in the presence of diffusion | 161 Simulation results and comparison | 162 Summary and conclusion | 165 Monte Carlo methods for computing macromolecules properties and solving related problems | 166 Diffusion-limited reaction rate and other integral parameters | 167 Formulation of the problem | 167 Capacitance calculations | 170 Walk in subdomains and efficient simulation of Brownian motion exit points | 172 Monte Carlo algorithms for boundary-value conditions containing the normal derivative | 174 WOS algorithm for mixed boundary-value conditions | 174 Mean-value relation at a point on the boundary | 176 Construction of the algorithm and its convergence | 177 Continuity BVP | 179 Monte Carlo method | 180 Integral representation at a boundary point | 182 Estimate for the boundary value | 184 Construction of the algorithm and its convergence | 185 Computing macromolecule energy | 187 Mathematical model and computational results | 188

Bibliography | 193

ix

1 Introduction It is well known that the random walk methods for boundary value problems (BVPs) for high-dimensional domains with complex boundaries are quite efficient, especially if it is necessary to find the solution not in all the points of a grid but only in some marked points of interest. One of the most impressive features of the Monte Carlo methods is the possibility to calculate probabilistic characteristics of the solutions to BVPs with random parameters (random boundary functions, random sources and even random boundaries). Monte Carlo methods for solving partial differential equations (PDEs) are based on the following: (a) classical probabilistic representations in the form of Wiener or diffusion path integrals, (b) probabilistic interpretation of integral equations equivalent to the original BVP, which results in representations of the solutions as expectations over Markov chains. In approach (a), diffusion processes generated by the relevant differential operator are simulated using numerical methods for solving ordinary stochastic differential equations [29]. To achieve the desired accuracy, it is necessary to take the discretization step small enough, which results in long random trajectories simulated. For PDEs with constant coefficients, however, it is possible to use the strong Markov property of the Wiener process and create much more efficient algorithms first constructed for the Laplace equation [76] known as the walk-on-spheres (WOS) method. This algorithm was later justified in the framework of approach (b) by passing to an equivalent integral equation of the second kind with generalized kernel and using the Markov chain that ‘solves’ this equation. This approach was developed for general second-order scalar elliptic equations, high-order equations related to the Laplace operator and some elliptic systems [98]. We now shortly present two different approaches for constructing and justifying the WOS algorithm: the first, conventional, coming from approach (a), and the second based on a converse mean value relation. Let us start with a simple case, the Dirichlet problem for the Laplace equation in a bounded domain G ⊂ R3 : ∆u(x) = 0, x ∈ G,

(1.1)

u(y) = φ(y), y ∈ Γ = ∂G.

(1.2)

We seek a regular solution to (1.1) and (1.2), i.e., u ∈ C2 (G)



C(G



Γ).

2 | 1 Introduction Let d* := supx∈G d(x), where d(x) is the largest radius of a sphere S(x, d(x)) centred at x and contained in G. Let w x (t) be the Wiener process starting at the point x ∈ G, and let τ Γ be the first passage time (the time of the first intersection of the process w x (t) with the boundary Γ). We suppose that the boundary Γ is regular so that (1.1) and (1.2) have the unique solution. Then [19], u(x) = Ex φ(w x (τ Γ )).

(1.3)

Note that in (1.3), only the random points on the boundary are involved. We thus can formulate the following problem: how to find these points without explicit simulation of the Wiener process inside the domain G? This problem was solved in the study of Müller [76] using the following considerations. In the sphere S(x, d(x)), representation (1.3) gives u(x) = Ex u(w x (τ S(x,d(x)) )).

(1.4)

The same representation is valid for all points y ∈ S(x, d(x)), so we can use the strong Markov property and write the conditional expectation: u(x) = Ex {Ey u(w y (τ S(y,d(y)) ))|x = w(0); y = w(τ S(x,d(x)) )}.

(1.5)

We can iterate this representation many times and remark that only random points lying on the spheres S(x, d(x)), S(y, d(y)), . . ., are involved. It is well known that the points w x (τ S(x,d(x)) ) are uniformly distributed over the sphere S(x, d(x)). Thus, we came to the definition of the WOS process starting at x: it is defined as the homogeneous Markov chain WS = WS{x0 , x1 , . . . , x k , . . .} such that x0 = x and x k = x k−1 + d(x k−1 )ω k ,

k = 1, 2, . . . ,

where {ω k } is a sequence of independent unit isotropic vectors. It is known [76] that x k → y ∈ Γ as k → ∞; however, the number of steps in the Markov chain WS is infinite with probability 1. Therefore, an ε-spherical process is introduced as follows. Let N ε = inf {n : d(x n ) ≤ ε}, then the ε-spherical process WSε = {x0 , x1 , . . . , x N ε } is obtained from WS by stopping after N ε steps. Let x¯ N ε be any point on S(x N ε , d(x N ε )) ∩ Γ, then we set ξˆ (x) = u(x N ε ),

ξ (x) = φ(x¯ N ε ).

(1.6)

The unbiasedness u(x) = Ex ξˆ (x) follows from (1.4), (1.5). An important question is what happens when ε → 0. We remark that E{u(x n+1 )|x1 , . . . , x n } = E{u(x n+1 )|x n } = u(x n )

1 Introduction

|

3

by the spherical mean value theorem. It means that {u(x0 ), . . . , u(x n ), . . . } is a martingale with respect to x1 , . . . , x n , . . .. Since {u(x0 ), . . . , u(x N ε )} is a stopped process constructed from {u(x0 ), . . . , u(x n ), . . . }, it is also a martingale with respect to x1 , . . . , x n [75]. Especially, E{ξˆ (x)|x1 , . . . , x k ; N ε ≥ k} = E{u(x N ε )|x k , N ε ≥ k) = u(x k ), and Ex ξˆ (x) = u(x). Let L = sup ||x − y||,

M = sup φ(x)

x,y∈G

x∈Γ

and assume that u(x) satisfies the condition |u(x) − u(y)| ≤ c||x − y ||,

x, y ∈ G.

It is not difficult to show [75] that Ex {(ξˆ (x) − u(x))2 } ≤ c2 L, and for the estimator ξ (x), the following estimation is true: Ex (ξ − u)2 ≤ c2 ε2 + 4Mcε + c2 L2 , and  Ex

1 ξ k (x) − u(x) n n

k=1

2

1 ≤ c2 ε2 + (4Mcε + c2 L2 ), n

where ξ n are independent ξ -estimators of u(x). We now turn to the second approach. It is well known that every solution to (1.1) satisfies the mean value relation:  1 u(x) = N r u(x) := u(x + rs) dΩ(s) (1.7) ωm S(x,r)

for each x ∈ G and for all spheres S(x, r) contained in G; ω m is the surface area of the unit sphere S(x, 1) in Rm . Besides, mean value relation (1.7) characterizes the solutions to (1.1). We need a stronger result, which presents an equivalent formulation of the problem (1.1), (1.2).

4 | 1 Introduction

Proposition 1.1 (Integral Formulation). We suppose that problem (1.1), (1.2) has a unique solution for any continuous function φ. Suppose that there exists a function  v ∈ C(G Γ), u|Γ = φ, v(x) = u(x) in Γ ε (ε ≥ 0) such that the mean value relation holds at every point x ∈ G \ Γ ε for the sphere S(x, d(x)). Then v(x) is the unique solution to the problem (1.1), (1.2). Proof. The proof uses the maximum property [12]. Since u also satisfies the mean value relation for every sphere contained in G, we conclude that the function u − v satisfies the mean value relation at every point x ∈ G \ Γ ε . Let F be a closed set of points x ∈ G \ Γ ε where u − v attains its maximum M. Let x0 be a point of F, which has the minimal distance from the surface Y = {y : d(y) = ε}. If x0 were an interior point of x ∈ G \ Γ ε , we could find a sphere S(x0 , r0 ) ⊂ G for which the mean value relation holds, thus u − v = M inside S(x0 , r0 ). Therefore, x0 should belong to Y. We repeat the same argument for the minimum of u − v. Since (u − v)|Y = 0, we have v ≡ u in G. Using this proposition, we rewrite an integral equation equivalent to the problem (1.1), (1.2). Let δ x (y) be a generalized density describing the uniform distribution on a sphere S(x, d(x)), and we define the kernel function as follows:  δ x (y) if x ∈ G \ Γ ε , (1.8) k ε (x, y) = 0 if x ∈ Γ ε . We define also a function f ε (x) in G:  f ε (x) =

0 u(x)

if x ∈ G \ Γ ε , if x ∈ Γ ε .

By the proposition, we can write an equivalent integral equation: u(x) = K ε u(x) + f ε (x), where the integral operator K ε is defined by  K ε ψ(x) = k ε (x, y)ψ(y)dy

(1.9)

(1.10)

G

for each ψ(x) ∈ C(G). Proposition 1.2. For any ε > 0, integral equation (1.9) has the unique solution given by u(x) = f ε (x) +

∞ 

K εi f ε (x).

i=1

Besides, this solution coincides with the solution to (1.1), (1.2).

(1.11)

1 Introduction

|

5

Proof. Let ε be fixed. To show the convergence of the series f ε (x) +

N 

K εi f ε (x),

(1.12)

i=1

it is sufficient to prove that K 2ε L∞ < 1 (this fact also implies the uniqueness of the solution to (1.11)). Let ν(ε) = ε2 /4(d* )2 . For x ∈ G \ Γ ε , we have ⎞ ⎛         k ε (x, y) k ε (y, y )dy dy = δ x (y)⎝ δ y (y )dy ⎠dy G

G

G

G\Γ ε



=

δ x (y)dy ≤ 1 − ν(ε) < 1.

G\Γ ε

Let v(x) := f ε (x) +

∞

i i=1 K ε f ε (x).

It is clear that v satisfies v(x) = K ε v(x) + f ε (x).

Now we remark that, to use (1.9) for numerical purposes, it is necessary to know the solution to the boundary problem in Γ ε . However, we obtain an approximation if we take in Γ ε u(x) = f¯ε (x) ≈ φ(x¯ ), x ∈ Γ ε ,

(1.13)



where x¯ is the point of Γ nearest to x, since u ∈ C(G Γ). Instead of (1.13), we could use any continuous extension of φ to Γ ε (the ideal case is the harmonic extension). Thus, let us consider the equation u ε (x) = K ε u ε (x) + f¯ε (x), x ∈ G \ Γ ε ,

(1.14)

where f¯ε is an approximation for u(x) in Γ ε , e.g. given by (1.13). Note that the solution to (1.14) is not harmonic, but it is clear that |u(x) − u ε (x)| = O(ω(ε)) as ε → 0, where ω(ε) is the continuity modulus of the function u(x) in Γ ε . For example, for Lipschitz continuous functions, we get |u(x) − u ε (x)| ≤ Cε.

Having (1.14) and the convergence of the Neumann series, it is possible to use the standard Monte Carlo estimators for the integral equations of the second kind [23]. If we choose the transitional density in each sphere in accordance with the kernel (i.e., uniformly on the surface of a sphere) and introduce absorption in Γ ε with probability 1 (no absorption in G \ Γ ε ), we exactly obtain the unbiased estimators ξˆ (x) and ξ (x). Estimations for the variance of the estimators of the solutions to the integral equations can be obtained from the analysis of the kernel k2ε /p, where p is a transitional density.

6 | 1 Introduction

This kernel in our case coincides with the kernel of the original integral equation, which leads to the convergence of the Neumann series representing the variance. Thus, we see that in the WOS algorithms, we have no necessity to simulate long trajectories of the Wiener process. Nevertheless, we have to construct a series of random points distributed inside the domain. However, we would like to find probabilistic representations in the form of an expectation taken over Markov chains defined on the boundary of the domain. We will see that this is really possible to do using the boundary integral equations of the potential theory and special Monte Carlo estimators for solving integral equations. We now shortly present the idea of the walk-on-boundary algorithm for solving (1.1), (1.2). Suppose for simplicity that the domain is convex and the solution can be represented in the form of a double-layer potential [120],  cos φ yx u(x) = μ(y)dσ(y), (1.15) 2π|x − y|2 Γ

where φ yx is the angle between the vector x − y and n y , the interior normal vector at the point y ∈ Γ, and μ(y) is a continuous function satisfying the boundary integral equation:  μ(y) = − p(y, y )μ(y )dσ(y ) + φ(y), (1.16) Γ

where p(y, y ) =

cos φ y y . 2π|y − y |2

It is clear that dΩ x =

dσ(y) cos φ yx |x − y |2

is a solid angle of view of the surface element dσ(y) from the point x ∈ G. Thus, the isotropic distribution of y in the angular measure Ω x at the point x corresponds to the distribution of y on Γ with the density p0 (x, y) =

cos φ yx . 4π|x − y|2

Since G is convex, p0 ≥ 0 and for arbitrary x ∈ G  p0 (x, y)dσ(y) = 1. Γ

(1.17)

1 Introduction

|

7

Thus, we can rewrite (1.15) in the form of the expectation u(x) = 2 Ex μ(Y0 ),

(1.18)

where Y0 is a random point on Γ, which is obtained as the intersection of the ray A x = {z : z = x + tω ; t ≥ 0} (ω is a unit isotropic vector), having random isotropic direction ω, with the boundary Γ. Note that if we could find μ(y) on Γ, we could use the representation (1.18) for numerical purposes. In Chapter 3, we shall see that μ(y) can be found from (1.16) by special iterations, which leads to the following algorithm. Let us define the following Markov chain on the boundary: WB x {Y0 , Y1 , . . . , Y m }, where the first point Y0 is obtained as described above, and Y n+1 = Y n + |Y n+1 − Y n |ω n , 3 where {ω k }m−1 k=0 is a sequence of independent unit isotropic vectors in R . Based on the process WB x {Y0 , Y1 , . . . , Y m }, we can construct the following random estimator (m ≥ 2):

ξ m (x) = 2[φ(Y0 ) − φ(Y1 ) + φ(Y2 ) + . . .] + (−1)m+1 φ(Y m ).

(1.19)

In Chapter 3, we will show that u(x) ≈ Eξ m , and the larger m, the higher is the accuracy  of this representation. More exactly, to

2 achieve the accuracy ε, the cost is O | lnε2ε| , which shows that this algorithm has a high efficiency. Remarkably, the same walk-on-boundary process can be used to solve interior and exterior Dirichlet, Neumann and the third BVPs. Note also that the method is grid free but gives the solution simultaneously in arbitrary points prescribed. When comparing with the classical probabilistic representations, there is no ε-error coming from the approximation in Γ ε . Note also that the parallelization of the walk-on-boundary algorithms is very simple, since the length of all the trajectories is one and the same. In this book, we construct and justify the walk-on-boundary algorithms for three classes of PDEs: (1) classical stationary potential problems (Chapter 3), (2) heat equation (Chapter 4) and (3) spatial elasticity problems (Chapter 5). The basic Monte Carlo methods using Markov chains for solving integral equations are presented in Chapter 2. In Chapter 6, we discuss different aspects related to the walk-on-boundary algorithms (the Robin problem and the ergodicity theorem, evaluation of derivatives, etc.).

2 Random walk algorithms for solving integral equations Conventional Monte Carlo methods for solving integral equations of the second kind are based on the Neumann series’ representation of the solution and, consequently, they are applicable only if simple iterations are convergent. This, in turn, happens when the spectral radius of the integral operator is less than 1 (more exact formulations are given in the next section). However, boundary integral equations of the potential theory cannot be treated in the framework of this conventional scheme, since the spectral radii of these equations are no less than 1. Therefore, we shall introduce different, in general biased estimators for solving integral equations whose Neumann series may be divergent. This technique can be applied to general integral equations with completely continuous operators.

2.1 Conventional Monte Carlo scheme Consider an integral equation of the second kind,  φ(x) = k(x, x )φ(x )dx + f (x),

(2.1)

Γ

or in the operator form, φ = Kφ + f . Here, Γ is a bounded domain in Rm . We suppose that the integral operator K is defined in a Banach space X(Γ) of functions integrable in the domain Γ. For example, if  – { |k(x, x )|r dx }1/r ≤ C1 for almost all x ∈ Γ, r > 0, Γ – { |k(x, x )|σ dx}1/σ ≤ C2 for almost all x ∈ Γ  , σ > 0, Γ

–

p > σ > p − r(p − 1),

then K is a completely continuous operator from L p (Γ) to L p (Γ) with the norm [47] ||K || ≤ C1−σ/p C2σ/p . 1

In particular, this is true for the kernels of the potential theory: k(x, x ) =

b(x, x ) , | x − x  |n

2.1 Conventional Monte Carlo scheme |

9

where b(x, x ) is a bounded continuous function, n < m. If, in addition, p > m/(m − n), then K : L p (Γ) → C(Γ). We now define a Markov chain in the phase space Γ as a sequence of random points Y = {Y0 , Y1 , . . . , Y n , . . .}, where the starting point Y0 is distributed in Γ with an initial density p0 (x), and the next points are determined by the transitional density p(Y i−1 → Y i ) = r(Y i−1 , Y i )(1 − g(Y i−1 )). Here r(Y i−1 , Y i ) is a conditional distribution density of Y i , Y i−1 fixed, and g(Y i−1 ) is the probability that the chain terminates in the state Y i−1 (i ≥ 1). Thus, depending on the value of g, the chain may have infinite or finite number, N, of states. Based on the Markov chain Y, we define the following random weights: the initial one Q0 =

f (Y0 ) , p0 (Y0 )

and Q i = Q i−1

k(Y i , Y i−1 ) , p(Y i−1 , Y i )

i ≥ 1.

It is supposed that p0 and p are consistent with f and k, respectively, i.e., p0 (x)  = 0 for {x : f (x)  = 0}, and p(x, y)  = 0 for {y : k(y, x)  = 0}. These suppositions are necessary for the weights Q i to be finite. Our goal is to evaluate a linear integral functional  I h = (φ, h) = φ(x)h(x)dx

(2.2)

Γ

for a function h ∈ X * (Γ). For example, if h = δ(x − x0 ), then I δ = φ(x0 ) is the solution to (2.1) at the point x0 . The following statement is true [23]. Proposition 2.1. Random variables ξ i = Q i h(Y i ) are unbiased estimators for (K i f , h). If the spectral radius of the integral operator  K1 φ = |k(x, y)|φ(y)dy Γ

(2.3)

10 | 2 Random walks for integral equations

is less than 1, then N 

Q i h(Y i )

(2.4)

Eξ i = (K i f , h),

(2.5)

Eξ = I h .

(2.6)

ξ=

i=0

is an unbiased estimator for I h ; i.e.,

Proof. Define δ n in the following way. We set δ n = 0 if the chain terminates at x n−1 , and δ n = 1 if the transition x n−1 → x n happens. Next, let ∆n =

n 

δk .

k=0

Thus, ∆ n = 1 until the first termination of the chain. Hence ξ=

∞ 

∆ n Q n h(x n ).

n=0

Suppose for a while that functions k(x , x), f and h are non-negative. Then, we can take expectation in the series termwise: Eξ =

∞ 

E[∆ n Q n h(x n )].

n=0

The conditional expectation formula gives E[∆ n Q n h(x n )] = E(Y0 ,...,Y n ) E [∆ n Q n h(x n )|Y0 , . . . , Y n ] = E(Y0 ,...,Y n ) [Q n h(Y n ) E (∆ n |Y0 , . . . , Y n )] = E(Y0 ,...,Y n ) [Q n h(Y n )  =

(1 − g(x n ))]

k=0

 dx0

n−1 

n−1 

dx n h(x n )p0 (x0 ) Γ

 n−1 f (x0 )  (1 − g(x k )) × p0 (x0 ) k=0

 =

k=0

n−1 

 dx0 . . .

Γ

= (K n f , h).

n−1   k(x , x ) k k+1 r(x k , x k+1 ) p(x k , x k+1 )

dx n f (x0 )h(x n ) Γ

k=0

k=0

 k(x k , x k+1 )

2.1 Conventional Monte Carlo scheme |

11

We have used here the obvious relation E[∆ n |Y0 , . . . , Y n ] = P(∆ n = 1|Y0 , . . . , Y n ) = P(δ0 = δ1 = . . . = δ n = 1|Y0 , . . . , Y n ) =

n−1 

[1 − g(x k )].

k=0

We turn now to the general case of functions k, f and h. Denote by Q(1) n the random weights constructed for k1 (x , x) = |k(x , x)|,

f1 (x) = |f (x)|,

h1 (x) = |h(x)|.

Then m  

|η m | = 

m    ∆ n Q n h(Y n )  ≤ |∆ n Q n h(Y n )|

n=0

=

m 

n=0

(1) ∆ n Q(1) n h 1 (Y n ) = η m ,

n=0

and η(1) m → ξ1 =

∞ 

∆ n Q(1) n h 1 (x n ).

n=0

By the assumptions made, Eξ1 = (φ1 , h1 ) < ∞ (here φ1 = K1 φ1 + f1 ). By the Lebesgue theorem, lim Eη m = E( lim η m ) = Eξ .

m→∞

m→∞

But Eη m = E

m 

∆ k Q k h(Y k ) =

k=0

m 

E(∆ k Q k h(Y k )) =

k=0

k=0

Thus, Eξ = lim Eη m = m→∞

∞ 

(K n f , h) = I h ,

n=0

since ||K n0 || ≤ ||K 1n0 || < 1

for some positive n0 .

m 

(K k f , h).

12 | 2 Random walks for integral equations

Note that it is possible to construct adjoint random estimators also. Let h(Y0 ) , p0 (Y0 ) k(Y i−1 , Y i ) , Q*i = Q*i−1 p(Y i−1 , Y i )

Q*0 =

i ≥ 1,

and ξ i* = Q*i f (Y i ), ξ* =

N 

Q*i f (Y i ).

(2.7)

i=0

Then, using the relations (K i f , h) = (f , K *i h), and (φ, h) = (f , φ* ), where φ* = K * φ* + h,  K * φ* (x) =

k(y, x)φ* (y)dy, Γ

we can prove that Eξ i* = Eξ i = (K i f , h),

(2.8)

Eξ * = Eξ = I h .

(2.9)

and

Taking f (x ) = δ(x − x ), p0 (x ) = δ(x − x ), Q0 = 1, we get φ* = h(x) + E

N 

Q n h(Y n ),

n=1

which can be proved by using the representation φ* =

∞  n=0

n

K * h.

(2.10)

2.1 Conventional Monte Carlo scheme

|

13

Or, considering the initial equation as an adjoint to φ* = K * φ* + h, we get φ = f +E

N 

Q*n f .

(2.11)

n=0

Variance of ξ can be represented as Dξ = (χ, [2φ* − h]) − I h2 , where χ is the Neumann series for the equation  2 f 2 (x) k (x, x )   )dx + χ(x . χ(x) = p(x , x) p0 (x)

(2.12)

(2.13)

Γ

Thus, if ||K1n0 || < 1 for some integer number n0 and the variance of the direct estimator for the integral equation with the kernel |k| is finite, then the variance Dξ is finite too. Remark 2.2. Let us consider a system of integral equations φ i (x) =

s 

k ij (x, y)φ j (y)dy + f i (x), i = 1, . . . , s

j=1

or in the matrix-operator form  K[x, y]Φ(y)dy + F(x).

Φ(x) = Γ

Suppose it is desired to find the linear functional  J(x) = P[x, y]Φ(y)dy ≡ (P, Φ). Γ

Here, K[x, y] is the matrix of kernels {K ij }si,j=1 , Φ = (φ1 , . . . , φ s )T , and P[x, y] is a given matrix. It is not difficult to extend the estimators ξ i , ξ i* for the iterations (P, K i [ ]F), where K[ ] is the matrix-integral operator generated by the matrix K[x, y]:  K[ ]F = K[x, y]F(y)dy. Γ

Indeed, let us choose a homogeneous Markov chain {Y0 , Y1 , . . .} defined by the initial density p0 and the transitional density p(x, y), such that p0 (y)  = 0 for y : P[x, y]Φ(y) ≡ 0,

14 | 2 Random walks for integral equations

and p(x, y)  = 0 for y : K[x, y]Φ(y) ≡ 0. Then the random variables ξ i = P[x, Y i ]Q i ,

ξ i* = Q*i F(Y i )

are unbiased estimators: Eξ i = Eξ i* = (P, K i [ ]F). Here, Q i are vector weights Qi =

K[Y i , Y i−1 ]Q i−1 , p(Y i−1 , Y i )

Q0 =

F(Y0 ) , p0 (Y0 )

K[Y i−1 , Y i ] , p(Y i−1 , Y i )

Q*0 =

P[x, Y0 ] . p0 (Y0 )

and Q*i are matrix weights: Q*i = Q*i−1

2.2 Biased estimators Let us consider an equation of type (2.1), introducing a complex variable λ: φ(x) = λKφ + f .

(2.14)

Let λ  = 0 be a non-singular value. This means there exists a unique solution to (2.14). Then, the operator R λ defined from I + λR λ = (I − λK)−1 is called a resolvent operator. Here, I denotes the identity operator. Let χ0 (K) be the set of characteristic numbers of the operator K, i.e., the set of such values of λ that the equation φ = λKφ has non-zero solutions. We suppose that K is a completely continuous operator. Then, the set χ0 (K) = {λ1 , λ2 , . . .} is countable and we assume that |λ1 | ≤ |λ2 | ≤ . . .. Without loss of generality, we suppose that it is necessary to find the solution to (2.14) at λ = λ* = 1 ∈ χ0 (K). If |λ1 | ≤ 1, then the Neumann series diverges at λ* = 1 and the conventional scheme described in Section 2.1 fails. However, the function R λ f is analytic in the circle {λ : |λ| < |λ1 |} and can be analytically continued to any regular λ, and, in particular, to the point λ* = 1. To carry out this continuation, we use the conformal mapping of the parameter λ [46].

2.2 Biased estimators

|

15

Let us take a simple connected subdomain D in the complex plane such that D ∩ χ0 (K) = ∅, and λ* ∈ D, 0 ∈ D. We choose a conformal mapping λ = ψ(η), ψ(0) = 0, which maps the unit disk ∆ = { η : | η | < 1}

(2.15)

onto D. Then η* = ψ−1 (λ* ) ∈ ∆, and ψ−1 (λ k ) ∈ ∆. We use now the expansion λi =

∞ 

k d(k) i η ,

k=1

where d(k) i =

 1  ∂k i [ψ(η)] . k! ∂η k η=0

Substitution of λ = ψ(η) into λR λ f gives F(η) = ψ(η)R ψ(η) f , and after the expansion into series F(η) =

∞ 

bk ηk ,

k=1

where bk =

k 

i d(k) i K f.

i=1

Thus, the series φ(x) = f (x) +

∞  k  k=1

 i k d(k) i K f (x) η *

(2.16a)

i=1

is absolutely and uniformly convergent. Let η0 = min |ψ−1 (λ k )| ≥ 1. k

We cut off the series, taking first n terms: φ(x) = f (x) +

n  i=1

i l(n) i K f (x) + ε(n; x),

(2.16b)

16 | 2 Random walks for integral equations

where l(n) i =

n 

k d(k) i η* ,

k=1



(2.17) n

 |η |  |ε(n; x)| ≤ const ·  *  . η0 We assume that the region D and the mapping ψ(η) can be chosen so that all the coefficients l(n) i and the value η * can be calculated with the desired accuracy. To evaluate I h , we use the unbiased estimators ξ i defined in (2.3) or ξ i* defined in (2.7). Then, we can introduce the random estimators ξ (n) =

n 

l(n) i Q i h(Y i ),

i=0

ξ * (n) =

n 

* l(n) i Q i f (Y i ).

(2.18)

i=0

By the construction, these estimators are δ-biased: I h = Eξ (n) + δ1 , I h = Eξ * (n) + δ2 ,

(2.19)

where δ1 = (ε(n; ·), h), δ2 = (f , ε* (n; ·)), l0(n) = 1, g(Y i ) = 0 for i < n and g(Y n ) = 1. Further, for ξ (n), we use the name ‘direct δ-estimator’, and for ξ * (n), we use the name ‘adjoint δ-estimator’ for I h .

2.3 Linear-fractional transformations and their relations to iterative processes Assume that a solution to the integral equation (2.14) with a completely continuous operator K can be represented in form (2.16a). We rewrite it as φ(x) = f (x) +

∞ 

φ k (x)η*k ,

(2.20)

k=1

where φ k (x) =

k  i=1

i d(k) i K f (x).

(2.21)

2.3 Linear-fractional transformations

|

17

Introduce the notation φ(m) (x) = f (x) +

m 

φ k (x)η*k .

(2.22)

k=1

Our goal is to study the iterative process for φ(m) generated by the linear fractional transformation (α, β are complex parameters): λ=

αη , 1 − βη

(2.23)

which maps the unit disk ∆ onto the half-plane:      α D = {λ : − |λ¯ − λ¯ 0 | < 0} = λ : |λ| < |β|λ + β

 α  , β

and λ0 = −

α|β| . β(1 + |β|)

In this case, i i−1 k−i , d(k) i = α C k−1 β

k ≥ i,

−1

η* = (α + β) . Note that (2.20) can be rewritten then as follows: φ(x) = f (x) +

∞ 

K1i f1 (x),

(2.24)

i=1

where K1 =

β α I+ K, α+β α+β

f1 =

α Kf . α+β

(2.25)

Consequently, we get φ0 (x) = αKf ,

φ k (x) = (βI + αK)φ k−1 (x), α φ (x) = K1 φ(m−1) (x) + f (x), α+β α Kf (x). φ(0) (x) = f (x) + α+β (m)

(2.26)

Thus, (2.24) shows that we can pass to a new integral equation with the integral operator K 1 in (2.25) and the right-hand side f1 . Assume that λ i (K) ∈ D. Then the series

18 | 2 Random walks for integral equations

in (2.24) converges. The respective unbiased estimators for I h are f (x0 )h(x0 )  Q k h(x k ), + p0 (x0 ) N

ξ=

k=0

f (x0 )h(x0 )  * * Q k f1 (x k ), + p0 (x0 ) N

ξ* =

(2.27)

k=0

where weights Q k , Q*k are determined as described above with changing K by K1 and f by f1 . The transitional density is taken in accordance with K1 . The upper number N in (2.27) can be taken fixed, or at random, if the Neumann series for K 1 converges. Here K 1 is the integral operator with the kernel |k1 |. The variance, however, may be unbounded. Remark 2.3. Note that we could derive the iterative process by the following transformation of the original integral equation. Taking a  = 0, we get φ = [(1 − a)I + aK]φ + af , where a = α/(α + β). The iterative process m→∞

φ (m) (x) → φ(x) is constructed as in (2.26), but with φ(0) (x) =

α f (x). α+β

Note also that the domain of convergence of the iterative process (2.26) is broader than D. Let, for example, β = 1, α > 0. Then D = {λ : λ > −α/2}. The Neumann series (2.24) converges if |ψ −1 (λ k )| > |η * |,

k = 1, 2, . . . ,

where {λ k } are the characteristic numbers of K. Let λ = x + iy, where x = λ and i is the imaginary unit. Then, we can rewrite this condition in the form

x−

1 2 2 1 + α 2 +y > . α+2 2+α

Consider now the transformation λ = α/(1− βη). It maps the disk ∆ onto the half-plane: D = {λ : |λ − α| < |β||λ|},

2.3 Linear-fractional transformations

|

19

and ψ(0)  = 0, i i−1 k d(k) i = α C i+k−1 β ,

η* = (1 − α)/β.

This corresponds to iterations of the resolvent operator: φ = (I − αK)−1 f +

∞  [(1 − α)(1 − αK)−1 ]i (I − αK)−1 αKf . i=1

Parameter α is chosen so that the operator (I − αK)−1 exists. In this case, we can use the transformation (α  = 1): φ = (1 − α)(I − αK)−1 φ + (I − αK)−1 αf . The iterative process generated by the transformation λ=

α 1 − βη

has the form φ(m) = (1 − α)(I − αK)−1 φ(m−1) + (I − αK)−1 αf ,

(2.28)

or φ(m) = αKφ(m) + (1 − α)φ(m−1) + αf . The domain of convergence of this iterative process is

x−

1 2 2 1 − α 2 +y > . 2−α 2−α

In this case, the Neumann series for the resolvent operator (1 − α)(I − αK)−1 converges. Consider now the iterative process (2.26), (2.28) for fixed m, N. In this case, it is possible to change the order of summation; hence, the approximate solution is represented as an operator polynomial (of finite degree) applied to the right-hand side of the equation. Thus, for the series (2.24), we get φ=

N 

a i (N)K i f + ε(N),

(2.29)

i=0

where a0 (N) = 1, a i (N) = a i

N−1  k=i−1

a=

α α+β ,

N fixed.

k−i+1 C i−1 , k (1 − a)

(2.30)

20 | 2 Random walks for integral equations

If we take a finite number of terms in the expansion of the resolvent operator [(1 − α)(I − αK)−1 ]n , then the iterative process (2.28) leads to the following expressions for the coefficients in (2.29): a i (N) = α i+1

N 

C ik (1 − α)k−1 , where N ≤ m − 1, or N ≥ m; i > N − m + 1,

k=1

and a i (N) = α i

i+m−1 

C ik (1 − α)k−i , where N ≥ m; i ≤ N − m + 1.

(2.31)

k=1

Note, however, that for a particular case of K, it is possible to construct a non-biased estimator for the kernel of the operator R, defined by (I − αK)−1 = I + αR. This makes possible the construction of estimators for φ using double randomization and substituting into the weights not the exact value of the kernel (I − αK)−1 but its unbiased estimator. This estimator has a finite variance if the order of singularity of the kernel of K is less than m/2, where m is the dimension of the problem. Consider a simple example, which will be used later on for construction of the walk-on-boundary algorithms. Assume that all the characteristic values are real, and λ k ∈ (−∞, −a), a > 0. Without loss of generality, we assume that (2.1) is to be solved at λ = λ* = 1. In this case, it is convenient to use for analytic continuation the following function that maps the disk ∆ onto the domain D a , the complex plane with a cut along the real axis from −a to −∞: λ = ψ(η) =

4aη . (1 − η)2

(2.32)

The series in the expansion of R ψ (η)f converges absolutely and uniformly on ∆, and the coefficients can be exactly calculated: k 2k−1 d(n) k = (4a) C k+n−1 .

To construct a biased random estimator, we choose such m that the remainder of the series is equal to a desired quantity ε. Then R λ* f ∼ =

m  k=1

b k η*k =

m  k=1

η*n

k  i=1

d(k) i ci =

m  k=1

c k l(m) k ,

(2.33)

2.3 Linear-fractional transformations

|

21

where l(m) k =

m 

n d(n) k η* .

(2.34)

n=k

The coefficients l(m) are calculated in advance, according to (2.34), and the integrals k c k = K k+1 f are calculated by Monte Carlo methods. Let ξ k be an unbiased Monte Carlo estimator for c k−1 . Hence, an ε-biased estimator for the solution to the integral equation has the form ζ ε(m) (x) = f (x) +

m 

ξ k l(m) k .

(2.35)

k=1

Theorem 2.1. For the transformation (2.32), l(m) ≤ 1 for all k and m. Let variances be k 2 bounded, Dξ k ≤ σ , and ε be the required order of the estimator ζ ε(m) error. Then, for ε → 0, the computational cost of this estimator is T ε = O(| ln(ε)|3 /ε2 ).

(2.36)

Proof. To prove the result, it is sufficient to show that l(m) ≤ 1. Note that k η* = ψ−1 (λ* ) = [(a + λ* )1/2 − a1/2 ]/[(a + λ* )1/2 + a1/2 ] > 0 (m) and d(n) = (4a)k c2k−1 k+n−1 > 0, n = k, k + 1, . . . Therefore, for a fixed arbitrary k, l k k (∞) monotonically increases as m increases. In the limit m → ∞, l k = 1, since (2.34) gives | ≤ 1 for arbitrary k and m. 1k , k = 1, 2, . . . From this, it follows that |l(m) k

The following statement establishes an analogous result for a particular class of transformations. Theorem 2.2. Assume that the conformal mapping λ = ψ(η) has only simple poles on the circle |η| = 1. Then, if D(k) ≡ Dξ k ≤ σ2 , and if q=

c|η* | < 1, 1 − |η* |

(2.37)

where c is the constant in the inequality |a n | ≤ c, a n = d1n , the cost of the estimator (2.35) is of order (| ln(ε)|4 /ε2 ). Proof. The estimate |a n | ≤ c can be obtained from the representation [83]: ∞  n=0

an zn =

k  i=1

 ci bn zn , + 1 − zi z ∞

n=0

22 | 2 Random walks for integral equations where |z i | = 1, i = 1, . . . , k, limn→∞ sup |b n |1/n < 1. From this, we get   k k      n |a n | =  ci zi + bn  ≤ |c i | + B, |b n | < B.   i=1

i=1

Therefore, |ψ(η* )| ≤ c

|η* | . 1 − |η* |

Using the Cauchy inequality, we obtain k |d (n) k |≤c

| η * |k , (1 − |η* |)k |η* |n

and consequently, |l (m) k |≤

m  c k | η * |k = q k (m − k + 1). (1 − |η* |)k n=k

Thus, Dζ ε(m) (x) ≤ m

M 

(m − k + 1)2 q2k D (k)
1, Dζ ε(m) < q2 σ2 m3 /(1 − |pq2 |) if |pq2 | < 1, and T ε = O(| ln(ε)|4 /ε2 ). Thus, to estimate the cost of (2.35), it is necessary to have more information about and D(k). l(m) k

2.4 Singular approximations

|

23

2.4 Asymptotically unbiased estimators based on singular approximations We now consider a different approach to solving the integral equation  φ(x) = k(x, y)φ(y)dy + f (x).

(2.38)

The algorithms considered here are based on the well-known method [46], which exploits a singular approximation to the kernel k(x, y). The solution to the equation to be solved is simply represented through linear functionals of auxiliary equations. This fact can be effectively exploited in the Monte Carlo method. The difference of the approach presented consists of the possibility to use a ‘few-point approximation’, in particular, a one-point approximation k(x, y) = k(x, y 0 )γ (y) + k1 (x, y).

(2.39)

It is required only that the Neumann series for the integral equation with the kernel |k1 (x, y)| converges. For the sake of simplicity, consider a system of linear algebraic equations, i.e., (2.38) of the form x = Ax + b, or in n coordinates xi =

n 

a ij x j + b i ,

i = 1, . . . , n.

(2.40)

d ij y j + a ij0 , i = 1, . . . , n

(2.41)

j=1

Consider two auxiliary linear systems zi =

n 

d ij z j + b i ,

j=1

yi =

n  j=1

where d ij = a ij − a ij0 γj are the entries of a matrix D, {a ij0 , i = 1, . . . , n} is a fixed column of the matrix A, γ T = (γ1 , . . . , γn ) is an arbitrary vector. Theorem 2.4. Assume that systems (2.40), (2.41) are uniquely solvable and (γ , y) =

n 

γi y i  = 1.

j=1

Then, n xi = zi + yi

j=1 γj z j

1−

n

j=1 γj y j

, i = 1, . . . , n.

(2.42)

24 | 2 Random walks for integral equations Proof. Let T = (γ , x). We prove first that x i = z i + y i T, i = 1, . . . , n. Indeed, zi + yi T =

n 

d ij z j + b i + T

j=1

n 

d ij y j + Ta ij0

j=1

= b i + a ij0 T +

n 

d ij (z j + y j T).

(2.43)

j=1

Substituting a ij = d ij + a ij0 γj into (2.40) yields xi =

n 

d ij x j + b i + a ij0 T,

j=1

so we obtain the desired equality. We use it in the following transformations: xi = zi + yi

n 

γj x j = z i + y i

j=1

= zi + yi

n 

n 

γj (z j + y j T)

j=1

γj z j + y i T(γ , y) = z i + y i (γ , z) + (x i − z i )(γ , y)

j=1

= z i (1 − (γ , y)) + y i (γ , z) + x i (γ , y), and arrive at the desired equality xi = zi +

y i (γ , z) , 1 − (γ , y)

which proves the theorem. The equality (2.42) will be useful if the solution of (2.41) is somehow simpler than the construction of the solution of the original equation. In particular, the situation when ϱ(A) > 1 but ϱ(D) < 1 is interesting from the point of view of Monte Carlo methods, since in this case, it is possible to construct estimators for z i , y i , (γ , z) and (γ , y) simultaneously, using a single Markov chain. Consider now the case when the kernel has the form (2.39). Then the auxiliary equations are  φ1 (x) = k1 (x, y)φ1 (y)dy + f (x), G

 φ2 (x) =

k1 (x, y)φ2 (y)dy + k(x, y0 ),

(2.44)

G

where y0 is an arbitrary fixed point in G; γ (y) is a function such that (γ , φ1 ) and (γ , φ2 ) exist.

2.4 Singular approximations

|

25

Theorem 2.5. Assume that equations (2.44) are uniquely solvable for arbitrary right-hand sides (from a class of functions), and suppose that  (γ , φ2 ) = γ (y)φ2 (y)dy  = 1. G

Then,  φ(x) = φ1 (x) + φ2 (x)

G

γ (y)φ 1 (y)dy

1 − (γ , φ2 )

.

(2.45)

Proof. The proof is analogous to that of Theorem 2.4. Indeed, substituting (2.39) into (2.38) yields  (2.46) φ(x) = f (x) + k1 (x, y)φ(y)dy + k(x, y 0 )J, G



where J = (γ , φ) = γ (y)φ(y)dy. We show now that φ(x) = φ1 (x) + φ2 (x)J. Indeed, G

 φ1 (x) + φ2 (x)J =

k1 (x, y)φ1 (y)dy + f (x) G



+J

k1 (x, y)φ2 (y)dy + k(x, y0 )J = f (x) + k(x, y0 )J G

 +

k1 (x, y)[φ1 (y) + φ2 (y)J]dy. G

Taking into account (2.46), we obtain the desired equality. Like in Theorem 2.4, we use it in the following transformations:  φ(x) = φ1 (x) + φ2 (x)J = φ1 (x) + φ2 (x) γ (y)φ(y)dy  = φ 1 (x) + φ2 (x)

G

γ (y)[φ 1 (y) + φ 2 (y)J]dy

= φ1 (x) + φ2 (x)(γ , φ1 ) + φ2 (x)J(γ , φ2 ) = φ1 (x) + φ2 (x)(γ , φ1 ) + [φ(x) − φ1 (x)](γ , φ2 ) = φ1 (x) + φ2 (x)(γ , φ1 ) + φ(x)(γ , φ2 ) − φ1 (x)(γ , φ2 ) = φ1 (x)[1 − (γ , φ2 )] + φ(x)(γ , φ2 ) + φ2 (x)(γ , φ1 ) and the theorem is proved. Using (2.45), it is possible to construct an asymptotically unbiased estimator for φ(x) provided that ϱ(K 1 ) < 1, where K 1 is the integral operator with the kernel |k1 |.

26 | 2 Random walks for integral equations

Let ξ1 (x), ξ2 (x), and ξ1γ (x), ξ1γ (x) be unbiased estimators for φ1 (x), φ2 (x), and (φ1 (x), γ ), (φ2 (x), γ ), respectively. Note that due to (2.44), all these estimators could be constructed using a single Markov chain; moreover, ξ1γ (x), ξ1γ (x) will differ only in the initial weights. Then, η(x) = ξ1 (x) + ξ2 (x) 1  (k)  ξ lγ , = ξ l(m) γ m

 ξ2(m) γ

1− ξ2(m) γ

,

m

l = 1, 2

(2.47)

k=1

, is an asymptotically unbiased estimator for φ(x). Indeed, let us assume that  ξ l(i) γ (i)  1 − ξ2γ , i = 1, 2, . . . , m are independent realizations of ξ1γ , 1 − ξ2γ , respectively, constructed using a Markov chain. Then, by the well-known theorem for distribution of a function of random variables [88], the random quantity ζ=

m 

ξ1(i)γ

i=1

 m 

−1 (1 − ξ2(i)γ )

i=1

is asymptotically Gaussian with the mean (γ , φ1 )/[1 − (γ , φ2 )] and the variance 2    cov[ξ1γ (1 − ξ2γ )] D(1 − ξ2γ ) Dξ1γ (γ , φ 1 ) 1 . − 2 + m 1 − (γ , φ2 ) (γ , φ1 )2 (γ , φ1 )[1 − (γ , φ2 )] [1 − (γ , φ2 )]2 Analogous arguments can be used for the random variable ξ2 (x) ξ1(m) /[1 −  ξ2(m) ]; hence, η is an asymptotically unbiased estimator for φ(x). γ γ The approach described above can be generalized to systems of integral equations. Systems of linear algebraic equations can also be treated. Consider a system of integral equations  φ(x) = K[x, y]dy + f (x), (2.48) G

where f (x) is a known vector; K[x, y] is a matrix with elements {k(x, y)}m i,j=1 , and φ(x) = (φ1 (x), . . . , φ m (x))T is the column vector of functions to be found. No special assumptions about the properties of the integral operator K[ ], the class of vector functions φ, f , and the domain G will be used. We assume only that the system (2.48) is uniquely solvable and generalize the representation (2.42). First, for simplicity, consider the case n = 1, i.e., M[x, y] = K[x, y] − δ(x)γ T (y),

2.4 Singular approximations

|

27

or in more detail M ij (x, y) = k ij (x, y) − δ i (x)γ i (y),

(2.49)

where δ(x) and γ (y) are some arbitrary column vectors with components δ1 (x), . . . , δ m (x) and γ 1 (y), . . . , γ m (y), respectively. We introduce two auxiliary systems of integral equations  ψ0 (x) = M[x, y]ψ0 (y)dy + f (x), G

 ψ1 (x) =

M[x, y]ψ1 (y)dy + δ(x).

(2.50)

G

Then, the following representation for the solution to (2.48) holds  T γ (y)ψ 0 (y)dy φ(x) = ψ0 (x) + ψ1 (x) G T . 1 − G γ (y)ψ1 (y)dy This relation is a corollary of the general Theorem 2.7 proved below. Let M[x, y] = K[x, y] −

n 

δ k (x)γkT (y).

(2.51)

k=1

Introduce n + 1 auxiliary systems of equations  ψ 0 (x) = M[x, y]ψ0 (y)dy + f (x),  ψ1 (x) = M[x, y]ψ1 (y)dy + δ1 (x),  ψ n (x) =

M[x, y]ψ n (y)dy + δ n (x).

(2.52)

Denote by A the matrix with elements  a ij = γiT (y)ψ j (y)dy and denote by b the vector with components  b i = γiT (y)ψ0 (y)dy,

i, j = 1, . . . , n.

Theorem 2.6. Assume that the system (2.52) is uniquely solvable and there exists an operator (I − A)−1 . Then the solution to (2.48) is represented as φ(x) = ψ0 + ψ T (x)J,

28 | 2 Random walks for integral equations

where the vector J is determined from the system of linear algebraic equations J = AJ + b.

(2.53)

Proof. Substituting K[x, y] from (2.51) into (2.48) yields   φ(x) = M[x, y]φ(y)dy + δ 1 (x) γ1T (y)φ(y)dy + · · ·  · · · + δ n (x) γnT (y)φ(y)dy + f (x)  =

M[x, y]φ(y)dy +

n 

δ i (x)J i + f (x),

(2.54)

i=1

where  Ji ≡

γiT (y)φ(y)dy =

  n

j

γi (y)φ j (y)dy.

j=1

Now, ψ0 (x) +

n 

 ψ i (x)J i =

n n     M[x, y] ψ0 (y) + ψ i (y)J i dy + δ i (x)J i + f (x). (2.55)

i=1

i=1

i=1

Comparison of (2.54) and (2.55) shows that φ(x) = ψ0 (x) +

n 

ψ i (x)J i .

i=1

Note that  Ji =

γiT (y)φ(y)dy =





γiT (y) ψ 0 (y) +

n 

 ψ j (y)J j dy,

j=1

or

 J i = J1

γiT (y)φ 1 (y)dy + · · · + J n



γiT (y)ψ n (y)dy +



γiT (y)ψ 0 (y)dy,

for i = 1, . . . , n, i.e., the vector J satisfies the system (2.53). Note that the approach described can be applied also to a system of linear equations x = Ax + b, where A is an m × m matrix. We introduce a matrix B = A − α1 β1T − . . . − α n β Tn ,

2.4 Singular approximations

| 29

where α1 , . . . , α n and β1 , . . . , β n are arbitrary column vectors, i.e., the matrix B is obtained from A by subtraction of singular matrices of the following form (n < m): ⎞ ⎛ 1 1 α i β i α1i β2i . . . α1i β ni ⎜α2 β1 α2 β2 . . . α2 β n ⎟ ⎜ i i i i⎟ α i β Ti = ⎜ i i ⎟. ⎝ ... ... ... ... ⎠ α ni β1i α ni β2i . . . α ni β ni Consider n + 1 auxiliary linear systems x0 = Bx0 + b, x1 = Bx1 + α1 , ... x n = Bx n + α n . As in the cases described above x = x0 +

n 

Jj xj ,

j=1

where J1 , . . . J m are components of the vector J, which satisfies the equation J = TJ + t. Here, T is the matrix with elements T ij = β Ti x j , and t is the vector with components t i = β Ti x0 . The results presented can be generalized to the case K(x, y) = M(x, y) +

n 

α i (x)β i (y).

i=1

Consider n + 1 auxiliary equations  φ0 (x) = M(x, y)φ0 (y)dy + f (x), G

 φ i (x) =

M(x, y)φ i (y)dy + α i (x),

i = 1, . . . , n.

G

Assuming that these equations are uniquely solvable, we derive, as previously, that φ(x) = φ0 (x) +

n  i=1

where J = (J1 , . . . , J n )T is determined from J = TJ + b,

φ i (x)J i ,

30 | 2 Random walks for integral equations

T being a matrix with elements  T ij = φ j (y)β i (y)dy, G

 bi =

φ0 (y)β i (y)dy,

i, j = 1, . . . , n,

G

provided that det T  = 0.

2.5 Integral equation of the first kind In this section, we consider the equation  k(x, y)φ(y)dy = f (x),

x∈Γ

(2.56)

Γ

or in the operator form Kφ = f , where K is a completely continuous, positive, self-adjoint operator: K : L 2 (Γ) → L2 (Γ). Here, Γ is a (m − 1)-dimensional hypersurface in Rm or a bounded domain in Rm−1 . Assuming that there exists a unique solution to (2.56), we construct a Monte Carlo estimator. We rewrite (2.56) in the form (κ  = 0) (for references see [98]) φ = (I − κK)φ + κf . It is known that φn ≡

n 

(I − κK)i κf → φ

i=0

as n → ∞, if 0 < κ < 2||K ||−1 . We represent φ n as a polynomial of K φn = κ

n 

i i C i+1 n+1 (−κ) K f .

i=0

Suppose that it is desired to find the solution to (2.56) at a point x ∈ Γ.

(2.57)

2.5 Integral equation of the first kind

| 31

We define a Markov chain, starting at x : Y0 = x, with a transitional density, p(Y i , Y i+1 ), consistent with the kernel k(Y i , Y i+1 ) of the integral operator K. Following Section 2.1, we can use the adjoint estimator for the iterations: K i f = EQ*i f (Y i ),

Q*0 = 1.

Consequently, ξ * (n; x) = κ

n 

i * C i+1 n+1 (−κ) Q i f (Y i )

(2.58)

i=0

is an unbiased estimator for φ n : φ n (x) = Eξ * (n; x). It is not difficult to estimate the error: 2(n+1) ||φ − φ n ||2 ≤ [1 − κλ−1 ||φ||2 + m]

∞ 

(φ, f k )2 ,

(2.59)

k=m+1

where λ1 ≤ λ2 ≤ . . . are the characteristic values of K, and f1 , f2 , . . . are the eigenfunctions. Suppose now that it is desired to calculate a linear functional I h = (φ, h) of the solution φ. Then (φ n , h) = Eξ n , where ξ n* =

ξ * (n; Y0 )h(Y0 ) . p0 (Y0 )

We can calculate the variance of this estimator: Dζ n* (n, x) =

n+1 

(C in )2 (−κ)2i d ii + 2

i=1

where

 K2i−1 (fK j−i f ),

d ij =  K2 f (x) =

n+1 

C jn C in (−κ)i+j d ij ,

j>i=1

h2 p0

 − (K i−1 f , h)(K j−1 f , h),

k2 (x, y) f (y)dy. p(x, y)

Γ

From this, we get Dζ * (n; x) ≤ max d ij (1 + κ)2n .

(2.60)

32 | 2 Random walks for integral equations

Consider now a different approach based on the iteration of the resolvent operator. We choose κ so that (I + κK)−1 exists. Then we can rewrite (2.56) in the form φ = (I + κK)−1 φ + κ(I + κK)−1 f . For κ > 0, φn =

n+1  [(I + κK)−1 ]i κf → φ

(2.61)

i=1

as n → ∞. We assume now that k(x, y) = O(|x − y|

1−m 2 +ε

)

as |x − y| → 0, ε > 0. Then we can construct an unbiased estimator (with a finite variance) for the resolvent operator R, defined by (I + κK)−1 = I − κR. The kernel r(x, y) of the operator R satisfies the equation (y is considered as a parameter): r(x, y) = −κKr(x, y) + k(x, y),

x ∈ Γ.

Hence, for 0 < κ < ||K ||−1 , we can define an estimator ζ (x, y) =

N 

Q*i,1 k(X i , y).

(2.62)

i=0

Here, {X i } is a terminating Markov chain with a transitional density l(X i−1 , X i ), X i ∈ Γ, X0 = x; N is the random number of states, and   k(X i−1 , X i ) . Q*0,1 = 1, Q*i,1 = Q*i−1,1 −κ l(X i−1 , X i ) Instead of exact values of r(y k−1 , y k ), we use their unbiased estimators ζ (y k−1 , y k ). From (2.61), we derive Eξ * (n, x) = φ n−1 (x), where ξ * (n; x) = nκf (x) + κ Q*0,2 = 1, Q*k,2 =

n 

k * C k+1 n+1 (−κ) Q k,2 f (Y k ),

k=1 Q*k−1,2 ζ (Y k−1 , Y k )/p(Y k−1 , Y k ).

(2.63)

2.5 Integral equation of the first kind

|

33

In the general case, however, it is not possible to construct a random estimator for r(x, y) with a finite variance. Then, we choose κ : 0 < κ < ||K ||−1 and take N terms in the representation (I + κK)−1 =

∞ 

C ki+k−1 (−κK)k .

k=0

Next, we substitute the resulting polynomial into (2.61) and arrive at the estimator: ξ * (n; x) = (n + 1)κf + κ

N 

* (−κ)k C k+1 n+1 Q k f (Y k ).

(2.64)

k=1

It is also possible to use a vector random estimator. Denote in (2.61) χ i = φ i−1 , i = 1, . . . , n and rewrite it in the form χ1 = −κKχ1 + κf , χ i = χ i−1 − κKχ i , i = 1, 2, . . . , n, or in a matrix-operator form χ = K χ + F, where χ = (χ1 , . . . χ n )T . We choose κ so that the spectral radius ρ(κK) is less than 1. Then, the Neumann series for K is convergent, and we can use the approximation φ n−1 =

n 

χi .

i=1

Hence, we can take φ(N) n−1 =

n  N  i=1

k=0

Kk F

 i

,

where the subscript ‘i’ shows that the ith component of the vector is taken. Since n and N are finite, for N ≤ n − 1, we get φ(N) n−1 =

N  (I − κK)k κf . k=0

This implies that φ(N) n−1 = φ N in (2.57). We make now some remarks about the computational cost of the estimators proposed. First of all, this cost depends on the rate of convergence of the Neumann series for the operator I − κK. If the set of characteristic numbers of K is bounded,

34 | 2 Random walks for integral equations then there exists such q ∈ (0, 1) that ||φ − φ n || < cq n .

It is reasonable to adjust the statistical error to the bias of ξ * . This ensures the consistency when calculating φ(x) by averaging over N independent samples of ξ * . Let ε = Cq n =

(1 + κ)2n 1/2 N

.

Then, n = ln ε/ ln q + C1 , 

N = C2 ε

2



ln(1+κ) ln q −1

.

If q = 1 − α, α  1, then the cost of the estimator ξ * is 

1 T = N · n = const − ln ε + C1 ε−2−2α ln(1+κ) . α Assume that λ m = O(m) (e.g. this is true in the case of the single-layer potential for convex Γ). Then, n = const/ε and T = O(ε −2 exp{ε−1 · C · ln(1 + κ)}).

3 Random walk-on-boundary algorithms for the Laplace equation 3.1 Newton potentials and boundary integral equations of the electrostatics In this chapter we consider classical BVPs of the potential theory in the stationary case. It means that we are concerned with the Laplace equation ∆u(x) = 0,

x ∈ G,

where function u is defined in some region G of the Euclidean space Rm and has continuous derivatives of at least second order. Let G be a bounded simple connected domain with a simple connected boundary Γ = ∂G. We denote G1 = Rm \ G, G = G ∪ Γ. From here on, m is considered to be greater than or equal to 3, which makes it possible to write down the general expressions of the formulas, but it must be mentioned that the two-dimensional (2D) case can be treated in the same way. We pass on to the definition of the surface potentials now. Suppose at first that Γ is a smooth Lyapunov surface [34]. It means that (1) at every point y ∈ Γ, there exists the definite normal vector n(y); (2) this vector, considered as a function of the point y on the surface Γ, is Hölder continuous, that is if x, y ∈ Γ, then |n(y) − n(x)| ≤ A |x − y |α

for some particular constants A and α ∈ (0, 1]; (3) there exists such constant d > 0 that if we consider the ball S(y, d) with the centre at some point y ∈ Γ and the radius equal to d, then the straight lines parallel to n(y) intersect Γ in S(y, d) only once. Let σ be a standard surface measure on Γ and μ(y), y ∈ Γ− some continuous function. Then, we can introduce the following functions (see for example [120]): function  2 |x − y |2−m μ(y)dσ(y) ≡ V[μ](x), (3.1) V(x) = (m − 2)σ m Γ

which is given the name single-layer potential and function  2 ∂ W(x) = |x − y |2−m μ(y)dσ(y) (m − 2)σ m ∂n(y) Γ  2 cos φ yx = μ(y)dσ(y) ≡ W[μ](x), σ m |x − y|m−1 Γ

(3.2)

36 | 3 Walk on Boundary for Laplace equation

which is named double-layer potential. We denote here m

σm =

2π 2 , Γ( m 2)

the surface area of a unit sphere in Rm , Γ(·) is the standard Γ− function. Hereafter we shall assume that n(y) is the internal (inward pointing) normal , where (.,.) is vector, φ yx is the angle between vectors n(y) and x − y, cos φ yx = (n(y),x−y) |x−y| a scalar product of two vectors in Rm . Properties of potentials V and W are thoroughly studied (see, for example, [34, 120]). First of all, V(x) and W(x) are harmonic functions everywhere in Rm \ Γ even for a piecewise smooth surface Γ. Theorem 3.1. If Γ is a Lyapunov surface and μ is a continuous function, then (1) V ∈ C(Rm ) (and is continuous on Γ consequently);  

∂V ∂V (2) there exist regular normal derivatives ∂n(x) and ∂n(x) on Γ, and +

−



∂V  ∂V 2 cos φ xy = ±μ(x) + μ(y)dσ(y). = ±μ(x) + σ m |x − y|m−1 ∂n(x) ± ∂n(x)

(3.3)

Γ

We remind that function u(x) is said to have regular normal derivative on Γ if, uniformly in all x ∈ Γ, there exists

∂u(x)  ∂u(x )  ≡ lim ( ∇ u(x ), n(x)) = , x ∈ ∓n(x), →x ∂n(x) x →x ∂n(x) ±

lim 

x

that is (·)+ denotes the limit from the exterior and (·)− denotes the limit from the interior ∂V is named the direct value of the normal derivative of of the domain G. Note that ∂n(x) a single-layer potential and it is continuous on Γ. An analogous theorem is valid for the double-layer potential. Theorem 3.2. If Γ is a Lyapunov surface and μ is a continuous function, then (1) W ∈ C(G) and C(G1 ); (2) limiting values of W satisfy the following relation:  2 cos φ yx μ(y)dσ(y), (W(x))± = ∓μ(x) + W(x) = ∓μ(x) + σ m |y − x|m−1

(3.4)

Γ

where W(x) is the direct value of a double-layer potential, and this function is continuous on Γ. These two formulas ((3.3) and (3.4)) will be widely used in this chapter. They serve as a foundation for the integral equations construction and, as a consequence, Monte Carlo estimators for solutions of the basic BVPs for the Laplace equation. In what follows, we pass on to the concrete classes of problems.

3.2 Interior Dirichlet problem

|

37

3.2 The interior Dirichlet problem and isotropic random walk-on-boundary process Consider the following problem: ∆u(x) = 0,

x∈G

u(y) = g(y),

y∈Γ

(3.5)

for some continuous function g. It is worth noting, that the last equality means, that the interior limit (u)− of the function u is equal to g. If the domain G and its boundary Γ satisfy the above-mentioned conditions, then the following statement is valid [120]. Theorem 3.3. The interior Dirichlet problem (3.5) is solvable for all continuous g, and its solution can be represented in the form of a double-layer potential (3.2). So, in the conditions of this theorem, we seek a solution of the problem (3.5) in the form of a double-layer potential W with an unknown density μ. As a consequence of relation (3.4), we come to the integral equation of the second kind for this function:  (3.6) μ(y) = λ k(y, y )μ(y )dσ(y ) + g(y), y ∈ Γ, Γ

where λ = 1, and the kernel of the integral operator K of this equation is equal to k(y, y ) = −

cos φ y y 2 · . σ m |y − y |m−1

(3.7)

On Lyapunov surfaces, this kernel is weakly singular [120]. Therefore, the operator K is compact [47] and, as a consequence of its compactness, all the poles of the resolvent of the integral equation (3.6) coincide with the characteristic values of this equation. The set of these numbers is at most countable, and we can enumerate them in accordance with their moduli: |λ1 | ≤ |λ2 | ≤ . . .. To construct a Monte Carlo estimator for the solution of the integral equation (3.6) and hence for the problem (3.5), we have to know the particular spectral properties of the equation (3.6). Simple but effective technique of the Green’s formulas provides a possibility to show that [32, 34] (1) all characteristic values are real numbers, and |λ i | ≥ 1; (2) they are simple poles of the resolvent; (3) λ1 = −1 is the characteristic value with the least modulus; (4) λ = 1 is not a characteristic value. The last property means that there exists the unique solution of the integral equation (3.6) for λ = 1. However, due to the third property, the Neumann series does not converge for an arbitrary function g. This is exactly the case, which has been considered in Chapter 2. Thus, we can follow the procedure that has been written out there.

38 | 3 Walk on Boundary for Laplace equation

Note first of all that by setting u(x) = W(x), we represent the solution in the form of an integral functional u(x) = (h x , μ), where μ is a solution of the integral equation (3.6) and the weight function is defined as follows: h x (y) =

cos φ yx 2 · . σ m |x − y|m−1

(3.8)

To begin with, let G be convex. In this case, h x > 0 for all x ∈ G and h x (y0 ) = 2p0 (y0 ) for x0 = x, where p0 (y0 ) =

cos φ y0 x0 1 · σ m |x0 − y0 |m−1

(3.9)

is the probability density function corresponding to the distribution of a point y on Γ uniform in a solid angle with its vertex at point x0 . This fact follows from a simple geometric consideration (see [34] as an example) that p0 (y)dσ(y) = dω(y) is an elementary solid angle. In the full analogy, we have that p(y i , y i+1 ) =

cos φ y i+1 y i 2 · σ m |y i − y i+1 |m−1

(3.10)

is the probability density function of the uniform distribution of a point y i+1 on Γ in a solid angle with its vertex at a point y i , which is also on Γ. Definition 3.1. Markov chain {y0 , y1 , . . .} of points y i on Γ is named the process of isotropic random walk on boundary (WB-process) if its initial point is distributed with the probability density p0 (y0 ) and its transitional density is p(y i , y i+1 ). To construct a trajectory of an isotropic WB-process, we choose some point x in the interior of G and then sample a ray with a random uniformly distributed direction coming out from this point. The unique in the case of convex Γ intersection of this ray with the boundary gives us the starting point y0 of the Markov chain. The analogous procedure is used to find the next point y i+1 when the previous y i is known. The only difference is that in this case, the direction y i+1 − y i is uniformly distributed in a half-space determined by the tangent plane at the point y i . It should be noted that we do not need to know the real positioning of the tangent plane since an isotropic distribution does not depend on the coordinate system. As a consequence, we can proceed as follows. Choose randomly a straight line (not a ray) with an isotropic direction in a fixed coordinate system. This line, as opposed to a ray, always intersects boundary Γ in the unique not equal to y i point, which we choose as y i+1 . Now we turn to definition of an estimator for the solution of BVP (3.5). It follows from (3.7) to (3.10) that the kernel of the integral equation (3.6) k(y, y ) is consistent with the transitional density p(y, y ) of the isotropic WB-process, and more than that,

3.2 Interior Dirichlet problem

|

39

|k(y, y )| = p(y, y ). So it is natural to use adjoint estimators (Chapter 2) for iterations

of the integral operator K and hence for a solution of BVP (3.5). As a result, we have Q*0 = 2, Q*i = 2 · (−1)i , and ξ * (x; n) =

n 

i 2 l(n) i (−1) g(y i )

(3.11)

i=0

are is δ(x; n)-biased estimator for u(x), where |δ(x; n)| ≤ const · q n , and q < 1; l(n) i defined by the concrete method of analytical continuation of the resolvent (Chapter 2). In this particular case, the characteristic set χ(K) is well-defined, and we can use comparatively simple domains D in the complex plane λ with the required properties. As an example, we put D = {λ : λ > −1}, i.e., the half-plane, and take the function λ = 2η/(1 − η), which maps the  unit  disk onto D. In this case,  λi  * * η = 1/3, q = η /η0 , η0 = min λ i +2  and l(n) i =

n 

2i (1/3)k C i−1 k−1 .

k=i

Another appropriate function is λ = 4η/(1 − η)2 . It maps the unit disk onto the whole plane, with the ray of real numbers  less than  to −1 excluded. Here, or equal √   η* = 3 − 2 2, q = η* /η0 , η0 = min1 + λ2i − λ2i 1 + λ i  and l(n) i =

n 

√

4i (3 − 2 2)k C2i−1 i+k−1 .

k=i

Remember now that λ1 = −1 is a simple pole of the resolvent, R, and |λ2 | > 1. Thus, R (λ) = (λ + 1)R λ is an analytical function of λ in the interior of the disk {λ : |λ| < |λ2 |}. As a consequence, this function can be expanded in the convergent operator series R (λ) = K + λ(K + K 2 ) + λ2 (K 2 + K 3 ) + . . . , and since R1 = 12 R (1), we can proceed as follows. Having taken a finite number of terms, we can evaluate the remainder using the fact that the series for R converges at the speed of a geometric progression with multiplier q ≤ |λ12 | . So we have μ(y) =

n−1  i=0

K i g(y) +

1 n K g(y) + ε(n; y), 2

40 | 3 Walk on Boundary for Laplace equation

and the estimator based on this representation is (3.11), where i = 1, . . . , n − 1, l(n) i = 1, 1 l(n) n = . 2 In Chapter 2, we showed that the resolvent analytical continuation and other methods of rearrangement of the original integral equation provide a possibility to construct new iterative algorithms for computing the solution. Often, such iterative procedure may be considered as a summation of Neumann series for a different integral equation. αη gives birth to a class of alternative In particular, linear fractional mapping λ = 1−βη representations μ=

∞ 

K1i g1 ,

i=0

where β α I+ K, α+β α+β α g. g1 = α+β

K1 =

Note that we cannot use the standard Monte Carlo procedure of constructing an estimator based on simulating trajectories of a terminating Markov chain. The reason is that the Neumann series for K 1 does not converge. So we take, as usual, a finite number of terms, n. This number is determined by the required accuracy, i.e., ||μ −

n 

K1i g1 || ≤ ε.

i=0

In our particular case, we take β = 1, 0 < α ≤ 2 and construct a special Markov chain with a transitional density, which is chosen in accordance with the kernel of the integral operator K1 . The initial point of this chain is sampled according to the density p0 from (3.9). Next, if a point y i is defined, we randomize the choice of the point y i+1 in the following way. We set y i+1 = y i with probability 1/(1 + α), and with probability α/(1+ α), we choose y i+1 uniformly distributed in the solid angle of view from the point y i . Thus, we have the following estimator: α  * Q i g(y i ), 1+α n

ξ α* (x; n) =

i=0

where Q*0 = 2, Q*i+1 = Q*i · b(y i , y i+1 ), and  1, with probability 1/(1 + α) . b(y i , y i+1 ) = −1, with probability α/(1 + α)

(3.12)

3.2 Interior Dirichlet problem

| 41

It is worth noting that multiplication of the resolvent by the term 1 + λ leads to some iterative method too, namely,  i 1 K g1 , g+ 2 ∞

μ=

i=0

1 2 (Kg + g), and we have to sum the Neumann series for the original operator

where g1 = K. These series converge here, since g1 is orthogonal to μ0 , the Robin’s potential density. It means that (μ0 , g1 ) = 0, and μ0 = −K * μ0 is the characteristic function (and the eigenfunction in this case), corresponding to λ1 = −1, the characteristic value of the adjoint integral operator K * . Lacking the absolute convergence of the Neumann series, we have to take a finite number of terms and construct an estimator based on trajectories of an isotropic WB-process. The adjoint estimators for iterations of the integral operator K are natural to be used here. Therefore, we have to know values of the function g1 at the points of WB-process. The double randomization principle provides a possibility to use unbiased conditionally independent estimators instead of exact values of g1 (y i ). As a result, we come to the following expression:  1 * 1  * k(y i , Y i ) Q i g(y i ) + g(Y i ) , Q0 g(y0 ) + 2 2 p(y i , Y i ) n

ξ0* (x; n) =

(3.13)

i=0

where Y i is sampled isotropically in accordance with the density p and is independent of the point y i+1 . We have Q*i = 2 · (−1)i , since we consider the domain G to be convex here. Recall now that everywhere in this chapter, we have supposed that the point x ∈ G, at which the solution of the BVP is computed, coincides with the point x0 , which is involved in the definition of the density function p0 (y0 ) (3.9). However, such requirement is not essential. For estimate to have finite mathematical expectation, it is sufficient to suppose that the ratio Q*0 (x, y0 ; x0 ) =

cos φ y0 x · |x0 − y0 |m−1 h x (y0 ) =2 cos φ y0 x0 · |x − y0 |m−1 p0 (y0 )

is bounded, or even less, that Q*0 is finite with probability one. As a consequence, solution u can be evaluated at several points x(j) ∈ G simultaneously. The only condition that should be satisfied is these points must be separated from the boundary (it means that a distance between x j and Γ is uniformly in j bounded away from zero). Then, Q*0 (x(j) , y0 ; x0 ) is uniformly bounded. The adjoint estimator that can be used here differs from (3.11) in its initial weight only: ξ * (x(j) ; n) =

n 

i Q*0 (x(j) , y0 ; x0 )l(n) i (−1) g(y i ).

(3.14)

i=0

Now we turn our attention to the problem of evaluating the derivatives of the solution. This task is often more important than solving the BVP itself. Let x be an interior point

42 | 3 Walk on Boundary for Laplace equation

in G. Analyticity of a double-layer potential W(x) at this point provides a possibility to differentiate this function under the integral sign. So we have

∂h  ∂u x (x) = , μ , k = 1, . . . , m, ∂x k ∂x k where ∂h x 2 1 x −y  n k (y) − m · cos φ yx · k k , (y) = · m ∂x k σ m |x − y| |x − y| x k , y k , n k (y) are coordinates of corresponding vectors x, y, n(y). It is clear that if we take a finite number of points x(j) in the interior of a convex domain G separated from its boundary Γ, the same estimator (3.12) can be used for computing not only the solution at these points but also all its first derivatives. The difference is contained in the weight Q*0 only, i.e., Q*0 (x(j) , y0 ; x0 ) =

∂h(j) x (y ) · (p0 (y0 ))−1 ∂x k 0

(3.15)

and Q*0 (x0 , y0 ; x0 ) =

n (y ) 2 x −y  k 0 − m 0k 0k |x 0 − y 0 | cos φ y0 x0 |x0 − y0 |

in the case x = x0 . It has been assumed so far that domain G is convex. Let us pass on to a more general case. The first consequence of a non-convexity is that function p(y i , y i+1 ) defined in (3.10) is not positive for all points y i+1 , and therefore, it cannot be used as a transitional density. From the geometric point of view, it means that a ray starting at the point y i (or at the point x0 ) can intersect Γ at several points, and in each of them, cos φ y i+1 y i has its own sign. The inherent relation between the kernel k and the isotropic random distribution of the next point of the Markov chain, the convenience of such random choice, leads to the desire of using the same p from (3.10) for constructing WB-process in a non-convex case. One of the possible ways to do this is following. Denote by q(y i , y i+1 ) ≥ 1 the number of intersections of an isotropically distributed straight line through the points y i , y i+1 , excluding y i . Then, with probability 1/q, we choose randomly one of these intersections and set it to be y i+1 . The transitional density that corresponds to such choice of the next point in the Markov chain is p(y i , y i+1 ) =

| cos φ y i+1 y i | 2 · . σ m q(y i , y i+1 ) · |y i − y i+1 |m−1

(3.16)

In the full analogy, the distribution density of the initial point is equal to p0 (y0 ) =

| cos φ y0 x0 | 1 · . σ m q(x0 , y0 ) · |x0 − y0 |m−1

(3.17)

3.3 Solution of the Neumann problem

|

43

A Markov chain constructed on a non-convex Γ in accordance with the densities p0 and p that are defined by formulas (3.17), (3.16) is also named an isotropic walk-on-boundary process. Having transformed the distribution densities that define WB-process, we inevitably come to changes in the form of estimators already constructed for the convex case. So we have ξ * (x; n) =

n 

i * l(n) i Q 0 (−1) T i g(y i ),

(3.18)

i=0

where Tj =

i−1 

q(y j , y j+1 ) · sign (cos φ y j+1 y j ),

T0 = 1.

(3.19)

j=0

In the full analogy with the convex case, the same trajectories of a WB-process can be used for simultaneous computation of a solution and its derivatives. We take the same estimator (3.17) changing the initial weight Q*0 only. Setting Q*0 =

h x (y0 ) 2 cos φ y0 x · |x0 − y0 |m−1 · q(x0 , y0 ) , = | cos φ y0 x0 | · |x − y 0 |m−1 p0 (y0 )

(3.20)

we obtain the estimator for a solution u(x). In particular, for x = x0 , Q*0 = 2q(x0 , y0 ) · sign(cos φ y0 x0 ). It is essential to note that in non-convex domains, the expression in (3.20) can be unbounded for x  = x0 . However, it does not mean that we cannot use WB-process for simultaneous computation of a solution u(x) at several different points. It is sufficient to change the distribution density p0 in such a way that p0 (y) ≥ c > 0 for nearly all y ∈ Γ. The same method of changing the initial distribution works in the case when we want to evaluate the derivatives of a solution and weights (3.15) are unbounded due to a non-convexity.

3.3 Solution of the Neumann problem Another classical well-posed problem is the Neumann BVP for the Laplace equation. It means that ∆u(x) = 0, x ∈ G1 ( or x ∈ G), ∂u (y) = g * (y), y ∈ Γ, ∂n

(3.21)

and u(x) → 0 for |x| → ∞ if we consider the exterior problem. The following theorem [120] is valid here.

44 | 3 Walk on Boundary for Laplace equation Theorem 3.4. The exterior Neumann problem (3.21) is solvable for all continuous g * and its solution can be represented in the form of a single-layer potential (3.1). As a consequence of this theorem, we can seek for a solution of the Neumann problem in the exterior domain G1 in the form of a single-layer potential with an unknown density μ* . This potential satisfies

the Laplace equation at any point x ∈ G1 . = g * and the formula (3.3) for the Combination of the boundary condition ∂V ∂n +

potential normal derivative makes it possible to write down the following integral equation of the second kind:  μ* (y) = λ k* (y, y )μ* (y )dσ(y ) + g * (y), (3.22) Γ

or in the operator form μ* = λK * μ* + g * , where λ = 1, and K * is the integral operator adjoint to the integral operator K from equation (3.6). Its kernel is k* (y, y ) = k(y , y) = −

cos φ yy 2 · . σ m |y − y |m−1

(3.23)

Spectral properties of the operator K are known. It means that we can easily obtain such properties for K * also. The characteristic set χ(K) lies on the real axis of a complex plane, and thus χ(K * ) = χ¯ (K) = χ(K), where λ¯ stands for a complex number conjugate to λ. It means that characteristic values of operators K and K * coincide, and we come to the possibility of using for integral equation (3.22) the same methods as for the integral equation (3.6). It is obvious that representation of a solution in the form of a single-layer potential can be considered as an integral functional u(x) = (μ* , h*x ), where h*x (y) =

2 1 · . (m − 2)σ m |x − y|m−2

This means that we find ourselves in a standard situation considered in Chapter 2 and, for the particular integral equation, in Section 3.2. As a consequence of (3.23), we come to a conclusion that an isotropic WB-process can be used here. Its transitional density p(y i , y i+1 ) is consistent with the kernel k* (y i+1 , y i ) and so it is natural to use a direct Monte Carlo estimator.

3.3 Solution of the Neumann problem

|

45

Let us begin with a convex case. In this instance |k* (y i+1 , y i )| = p(y i , y i+1 ),

Q i = (−1) · Q i−1 ,

and so the δ(x; n)-biased estimator for a solution of the Neumann problem (3.21) looks like as follows: ξ (x; n) =

n 

i * l(n) i Q 0 (−1) h x (y i ).

(3.24)

i=0

It has been mentioned already that the same transformations of an integral equation that have been used in the case of the interior Dirichlet problem to obtain a convergent series is appropriate for equation (3.22) also. As a result, the same analytical continuation methods and, hence, coefficients l(n) i can be used here. The main difference between the estimators (3.11) and (3.24) lies in the initial weight Q0 = g * (y0 )/p0 (y0 ). Recall now that the point x where the solution is estimated lies outside of G, and so there is no natural choice of a point x0 inside G that can be used for the density p0 definition (3.9). So the distribution of the initial point of WB-process is arbitrary to some extent. In a convex case, one of the options could be the isotropic density function (3.9), where x0 ∈ G is a suitably chosen point. It is always desired that an estimator is bounded with probability 1. In that instance, such condition is satisfied if the point x is separated from the boundary, Γ. Here, x ∈ G1 is the point, at which we want to find a solution. It is obvious that a solution of the exterior Neumann problem and its derivatives can be computed simultaneously using the same trajectories of a WB-process. The last assertion is true since a single-layer potential is an analytical function at the inner points of the domain. So ξ k (x; n) =

n 

i l(n) i Q 0 (−1)

i=0

2 y ik − x k · σ m | y i − x |m

∂u is an estimator for the derivative ∂x (x), where a point x = (x1 , x2 , . . . , x m ) is separated k from the boundary Γ. This expression is obtained from the formula (3.24), in which the function h*x (y i ) is substituted by its derivative ∂x∂ k h*x (y i ). In the full analogy with (3.12), (3.13), to construct a different estimator for the solution of BVP (3.21), we can use representations of μ* in the form of a Neumann series for a transformed integral equation. As a result, we have

μ* =

∞  (K1* )i g1* , i=0

46 | 3 Walk on Boundary for Laplace equation

where β α I+ K* , α+β α+β α * g1* = g . α+β

K1* =

To construct an estimator based on this representation, we proceed as follows. Let us take such a finite number of terms in the series that is sufficient to attain the desired accuracy. To do this, we use the information on the speed of convergence, which is αη determined by the linear fractional mapping λ = 1−βη . Next we have u(x) = (μ* , h*x ) =

n



 (K1* )i g1* , h*x + δ(x; n)

i=0

=

n 

(g1* , K1i h*x ) + δ(x; n).

i=0

So we see that the same Markov chain that has been defined in the process of the estimator (3.12) construction can be used in this case. Finally, we come to the following estimator (here β = 1, 0 < α ≤ 2): α  Q i h*x (y i ), 1+α n

ξ α (x; n) =

i=0

where Q0 =

g * (y0 ) , Q i+1 = Q i b(y i , y i+1 ) p0 (y0 )

for the same random variables b that were defined in (3.12). Consider now the method based on multiplication of the resolvent by the term 1 + λ. In the full analogy with the case of integral operator K, we come to the following convergent series for the operator K * : 1 *  * i * (K ) g1 , g + 2 ∞

μ* =

i=0

where g1* = (g * + K * g * )/2 is orthogonal to μ1 , the characteristic function of the operator (K * )* = K that corresponds to λ1 = −1. Here, μ1 is constant. Taking a finite number of terms in this series, we have u(x) = (μ* , h*x ) 1 * *  * i * * (K ) g1 , h x + δ(x; n) (g , h x ) + 2 n

=

i=0

3.3 Solution of the Neumann problem

|

47

1 * *  * i * (g1 , K h x ) + δ(x; n) (g , h x ) + 2 n

=

i=0

1 * * 1 * i * (g , K (h x + Kh*x )) + δ(x; n). (g , h x ) + 2 2 n

=

i=0

It is essential to note that the last transformation is valid for finite sum only. Using this expression we construct an estimator, which is based on simulating trajectories of an isotropic WB-process. In the full analogy with (3.13), the double randomization principle gives us a possibility to use unbiased conditionally independent estimators instead of exact values of Kh*x :  1 1 * k(y i , Y i ) * Q i h x (y i ) + h x (Y i ) . Q0 h*x (y0 ) + 2 2 p(y i , Y i ) n

ξ0 (x; n) =

i=0

Here, Y i is isotropically distributed with density p and is independent of the point y i+1 . In a convex case, we have Q i = (−1)i Q0 , Q0 =

g * (y0 ) k(y i , Y i ) , = −1. p0 (y0 ) p(y i , Y i )

We now turn our attention to a non-convex case. It is obvious that the same transformation of WB-process that has been used for the interior Dirichlet problem is applicable for the exterior Neumann problem also. So we can write out the basic estimator for u(x), a solution of (3.21): ξ (x; n) =

n 

i * l(n) i Q 0 (−1) T i h x (y i ),

(3.25)

i=0

where {y i } is an isotropic WB-process with the transitional density (3.16) and an * 0) initial density p0 , which is consistent with the boundary function g * , Q0 = pg (y and 0 (y 0 ) coefficients T i are defined in (3.19). Let us consider the interior Neumann problem now. It is well known that for arbitrary boundary values of the normal derivative, a solution of (3.21) for x ∈ G does not necessarily exist. Namely, the following proposition is valid [120]. Theorem 3.5. The interior Neumann problem is solvable for all continuous functions g * satisfying the orthogonality condition  g * (y)dσ(y) = 0. (3.26) Γ

Its solution can be represented in the form of a single-layer potential, and it is unique up to addition of an arbitrary constant function.

48 | 3 Walk on Boundary for Laplace equation

Thus, we consider BVP (3.21) in the interior of a domain G and seek its solution in * the form of a single-layer potential (3.1)  an unknown density μ . In that instance,

with = g * and so we can use Theorem 3.1 to derive the boundary condition means that ∂V ∂n −

an integral equation. From (3.3), we get  cos φ yy *  2 * · μ (y )dσ(y ) − g * (y) μ (y) = λ σ m |y − y |m−1

(3.27)

Γ

or μ* = λ(−K * )μ* − g * ,

λ = 1.

We see that the integral operator here differs from the integral operator K * , considered in the case of the exterior Neumann problem, only in its sign. From here, we have that λ1 = 1 is a characteristic value for the integral equation (3.27). The Fredholm theorems give some conditions of solvability of the equation (3.27) (see [47], for example). A solution exists and is unique if and only if the free term −g * of the equation (3.27) is orthogonal to all characteristic functions of the integral operator −K corresponding to the characteristic value λ1 = 1. It is well known that there exists only one such function and it is constant on Γ. So we see that (3.26) provides all necessary conditions of solvability and a general solution of (3.27) can be written out in the form μ* = μ*0 + cμ0 ,

(3.28)

where c is an arbitrary constant, μ0 is the Robin’s potential density and μ*0 is a particular solution of (3.27). It will be sought for in the form of a sum of the Neumann series for the equation (3.27). From (3.28), we get  2 μ*0 (y) + cμ0 (y) · dσ(y) = V0* (x) + const, u(x) = σm |x − y |m−1 Γ

since the Robin potential  V0 (x) =

μ0 (y) |x − y |m−1

dσ(y)

Γ

is constant in G. This property can be easily obtained from the definition of μ0 (it means that V0 is a solution of (3.21) with g * = 0) and the Green’s formulas [120]. We set V0 (x) equal to 1. It is obvious that the Neumann series does not absolutely converge here. Hence, to construct an estimator, we will use a finite number of iterations of the integral operator −K * . So we have

3.3 Solution of the Neumann problem

V0* (x) =

|

49

∞ 

 (−K * )i (−g * ), h*x i=0

n

  = (−K * )i (−g * ), h*x + δ(x; n) i=0 n

  = − g* , (−K)i h*x + δ(x; n),

(3.29)

i=0

n and λ2 is the second characteristic value of −K * . In a where |δ(x; n)| ≤ const |λ12 | convex case, the kernel of the integral operator −K is equal to the transition density p from (3.10). So, to construct an estimator for V0* (x), it is natural to use an isotropic WB-process. It is essential to note that to make the estimator effective, we have to take an advantage of the orthogonality condition (3.26). Divide Γ into two parts Γ = Γ+ ∪ Γ− , where Γ+ = {y ∈ Γ : g * (y) ≥ 0}, Γ− = {y ∈ Γ : g * (y) < 0}, and let p1 (y), p2 (y) be density functions on Γ+ , Γ− , respectively. Next, we sample two random points y10 and y20 distributed on Γ+ , Γ− with corresponding densities p1 , p2 and construct two trajectories of an isotropic WB-process starting from these points. This procedure of branching, which is generally called stratified sampling [23] or, for a special case, antithetic variates [36], leads to an estimator with less variance. So we have  * 1  n  g (y0 ) 1 g * (y20 ) 2 2 1 2−m 2 2−m , (3.30) | x − y | + | x − y | ξ (x; n) = − T T i i (m − 2)σ m p1 (y10 ) i p2 (y20 ) i i=0

where upper indices indicate that T ik are computed on the corresponding WB-process {y0k , y1k , . . . , y kn }, k = 1, 2. In a convex case, all T ik = 1, and we can simplify estimator (3.30) by sampling, as an example, y10 , y20 uniformly on Γ+ and Γ− , respectively. In  this case, p1 = p2 = 2g0−1 , where g0 = |g * (y)|dσ(y), and we arrive at a simplified form Γ

of the estimate: ξ (x; n) = −

n  i=0

! g0 |x − y 1i |2−m − |x − y 2i |2−m . (m − 2)σ m

WB-process trajectories do not depend on the point x ∈ G. As a consequence, we can compute a solution of the interior Neumann problem and its derivatives simultaneously at several points separated from the boundary. The last condition is sufficient for the estimator to have values bounded with probability 1. The same is

50 | 3 Walk on Boundary for Laplace equation

valid for the derivatives’ estimators:   n  g * (y20 ) 2 x k − y2ik 2 g * (y10 ) 1 x k − y1ik , + ξ k (x; n) = T T σ m p1 (y10 ) i |x − y1i |m p2 (y20 ) i |x − y1i |m i=0

which can be obtained by the direct differentiation of (3.30). So far we have supposed that we estimate a solution and its derivatives at a point x, which is separated from the boundary. In that case, kernels of potentials W(x), V(x) and their derivatives are uniformly bounded. The situation becomes essentially different when x tends to Γ. Some special methods for constructing effective estimators for the solution values at the points near the boundary are given in Section 3.7.

3.4 Random estimators for the exterior Dirichlet problem In this section, we consider the Dirichlet problem in G1 , which is exterior to a bounded domain G. It means that we want to find a harmonic function u: ∆u(x) = 0,

x ∈ G1 ,

(3.31)

which satisfies the boundary condition u(y) = g(y),

y ∈ Γ,

and the additional requirement, u(x) → 0 for |x| → ∞. Suppose first that u(x) is represented as a double-layer potential W[μ](x) with an unknown density μ. The relation (3.4) and the boundary condition (u)+ = g give rise to the integral equation  cos φ y y 2 μ(y) = λ · μ(y )dσ(y ) − g(y), (3.32) σ m |y − y |m−1 Γ

or μ = λ(−K)μ − g,

λ = 1.

For an arbitrary g, solution of this equation does not necessarily exist, since λ1 = 1 is the characteristic value of the integral operator −K. With the Fredholm theorems, we come to a conclusion that the integral equation (3.32) is solvable if and only if (μ0 , g) = 0, where μ0 is the Robin’s potential density. In that case, μ = μ(0) + c,

(3.33)

3.4 Exterior Dirichlet problem

|

51

where μ(0) is some particular solution of (3.21) (the sum of the Neumann series of (3.32), for example) and c, which is constant on Γ, is the unique eigenfunction of −K corresponding to λ1 = 1. As a consequence, we come to the following [120] statement. Proposition 3.2. The exterior Dirichlet problem has a unique solution in the form of a double-layer potential for all continuous functions g that satisfy the orthogonality condition (3.33). The uniqueness here follows from the Gauss formulas for a double-layer potential with constant density.  In particular, h x (y)dσ(y) = 0 when x ∈ G1 (Günter, 1934 [34]). Γ

In contrast to the analogous condition (3.26) for the case of the interior Neumann problem, the orthogonality condition (3.33) is not natural for the BVP considered here. It originates from the representation of a solution in the form of a double-layer potential. This representation allows us to find only such solutions that tend asymptotically to zero as 0(|x|1−m ) when |x| → ∞. However, in the original setting of the BVP, speed of convergence is not prescribed. It is well known that harmonic functions tend to zero as O(|x|2−m ) when |x| → ∞. Therefore, if the condition (3.33) is not satisfied, it means only that there is no such solution to the problem that can be represented in the form of a double-layer potential. Without loss of generality, we may suppose that 0 ∈ G. We seek a solution of the exterior Dirichlet problem in the form of a sum of a double-layer potential and a Newton potential α|x|2−m of an unknown charge α positioned at the point x = 0: u(x) = W[μ](x) +

α |x |m−2

.

(3.34)

Taking into consideration (3.4) and the boundary condition, we come to the integral equation μ(y) = (−K)μ(y) + f (y), where f (y) =

α |y |m−2

− g(y).

This equation differs from (3.32) in its free term only. So the same solvability condition must be satisfied, and here it takes the following form:   α − g(y) μ0 (y)dσ(y) = 0 (3.35) |y |m−2 Γ

or

 αV0 (0) −

g(y)μ0 (y)dσ(y) = 0, Γ

52 | 3 Walk on Boundary for Laplace equation where V0 is the Robin potential, which is equal to 1 since 0 ∈ G. So we have  α = g(y)μ0 (y)dσ(y),

(3.36)

Γ

and the following theorem is valid. Theorem 3.6. The exterior Dirichlet problem is solvable for all continuous functions g, and its solution can be represented as a sum of a double-layer potential and the potential  1 g(y)μ0 (y)dσ(y). |x |m−2 Γ

Note that α = 0 means the condition (3.33) is satisfied. So, the representation u(x) = W[μ](x) is a special case of a more general situation considered in this theorem. So we have the convergent Neumann series μ(y) =

∞  (−K)i f (y), i=0

since orthogonality condition (3.35) is satisfied. Lacking the absolute convergence, we have to take a finite number of terms to construct an estimator. As a consequence, we get n

  (−K)i f + δ(x; n), W(x) = (h x , μ) = h x , i=0

where |δ(x; n)| ≤ const

1 |λ2 |

n

.

In the full analogy with the interior Neumann problem, we have



h x (y)dσ(y) = 0

Γ

(see (3.29), (3.26)), and so we process in the same way. Divide Γ into two parts Γ = Γ+ ∪ Γ− , where Γ+ = {y ∈ Γ : h x (y) ≥ 0}, Γ− = {y ∈ Γ : h x (y) < 0}, and construct two trajectories of an isotropic WB-process, {y1i } and {y2i }, with starting points y10 distributed on Γ+ with density p10 and y20 distributed on Γ− with density p20 , respectively. These densities must be consistent with the function h x (y), which, as we

3.4 Exterior Dirichlet problem

|

53

know, has an apparent geometric interpretation. Namely, | cos φ yx | σm |h x (y)|dσ(y) = dσ(y) 2 |y − x |m−1

is a solid angle of view of a surface element dσ(y) from the point x. So we can proceed as follows. Choose an isotropically distributed direction of a ray coming out of the point x ∈ G1 . This ray intersects Γ with probability Ω/σ m , where  cos φ yx Ω(x) = dσ(y) (3.37) |y − x |m−1 q(x, y) Γ+

is a solid angle of view of the whole surface Γ from the point x. With a conditional probability 1, there will be even number of intersections. Denote this number by 2q(x, y). There will be q points on Γ+ and q points on Γ− . In a convex case, q = 1 and so we can take these intersections as the initial points y10 and y20 , respectively. If a boundary is not convex, there can arise such situations that q > 1 and in that case we choose randomly with probability q−1 one of the points on Γ+ and one on Γ− as y10 and y20 , respectively. As a result, we come to the following initial density: p1(2) 0 (x; y) =

| cos φ yx | . Ωq(x, y)|x − y|m−1

(3.38)

Having sampled initial points, we construct WB-processes {y1i } and {y2i } for i ≥ 1 using the transitional density (3.16). Since this density is consistent with k(y i+1 , y i ), an adjoint estimator is natural to be used here. Therefore, ξ * (x; n) = Q*0 (x)

n 

(T i1 f (y1i ) − T i2 f (y2i ))

(3.39)

i=0

is the desired δ-biased estimator for W(x). Coefficients T ik are calculated according to formula (3.19) using trajectories of the corresponding Markov chains {y ki }, k = 1, 2, Q*0 (x) =

h x (y10 ) Ω(x) . = 2q(x, y10 ) 1 1 σm p0 (y0 )

Two quantities are not defined yet, Ω and λ. Note that ξ * depends on Ω linearly and so we can use an unbiased estimator instead of its exact value. The geometric algorithm of sampling initial points gives us a clue to construction of this estimator. Here, we have the probability of success (i.e., of intersection) equal to Ω/σ m if the ray direction is sampled in the whole solid angle. Recall now that all the estimators constructed in this book are used in a common statistical way. It means that a large enough number, N, of independent realizations of ξ * is constructed, and then they are averaged in order to obtain the desired solution. Denote by N1 a number of independently sampled

54 | 3 Walk on Boundary for Laplace equation

random directions, N1 > N. Then N/N1 is an unbiased estimator for Ω/σ m . So, instead of Q*0 , we can use N . Q¯ *0 = 2q(x, y10 ) · N1 To compute the value of α, two limit theorems can be used. It is known that λ1 = 1 is the maximum eigenvalue of the integral operator −K * and μ0 is the unique eigenfunction corresponding to this value. So we have [32] μ0 (y) = lim (−K * )n z(y) · const n→∞

uniformly over y ∈ Γ for an arbitrary non-zero continuous function z. Using direct estimators constructed for an infinite isotropic WB-process {y i ; i = 0, 1, . . .}, we come to the following: α=

limn→∞ E(Q0 T n g(y n )) , limn→∞ E(Q0 T n |y n |2−m )

where Q0 =

z(y0 ) . p0 (y0 )

Another way of finding μ0 and integral functionals of this function follows from the ergodic theorem (Section 6.1). With probability 1, we have limn→∞ 1n α= limn→∞ 1n

n

i=1 n

g(y i ) .

|y i |2−m

i=1

From the computational point of view, it would be more convenient to rearrange (3.39) in such a way that the parameter α is used in the final expression only. Hence, " n  * (T i1 |y1i |2−m − T i2 |y2i |2−m )) ξ (x; n) = α Q¯ *0 (x) − Q¯ *0 (x)

i=0 n 

#

(T i1 g(y1i ) − T i2 g(y2i ))

.

(3.40)

i=0

It is essential to note that this estimator has finite variance even in the case when the point x approaches the boundary, Γ, since singularity of the function h x (y) is included in the distribution density of the initial point. As a consequence, in the neighbourhood of the boundary, a solution of the problem can be evaluated at one point only. Suppose now that points x0 , x1 , . . . , x r are separated from the boundary. Universal initial densities p10 , p20 from (3.38) cannot be used in this case since fraction

3.5 Third boundary value problem

|

55

h x j (y0k )(p0k (x0 ; y0k ))−1 is not uniformly bounded on Γ. Hence, the choice of special densities p0k , k = 1, 2 consistent with all h x j (y0k ) depends on the geometry of Γ.

3.5 Third BVP and alternative methods of solving the Dirichlet problem We consider here the third classical BVP for the Laplace equation ∆u(x) = 0, x ∈ G (or x ∈ G1 ), ∂u (y) − H(y)u(y) = g(y), y ∈ Γ, ∂n

(3.41)

where H ≡ 0 is some continuous function. Theorem 3.7. The third BVP is solvable for all continuous functions g, and its solution can be represented in the form of a single-layer potential, if H ≥ 0 for the interior problem and H ≤ 0 for the exterior one. So, let us suppose that a solution u(x) of the problem (3.41) equals V[μ](x), a single-layer potential with an unknown density μ(y). Theorem 3.1 gives us the integral equation of the second kind for this density:  (3.42) μ(y) = λ k*H (y, y )μ(y )dσ(y ) + f (y), y ∈ Γ, Γ

where k*H (y, y ) =

2 σm



 cos φ yy H(y) , − |y − y  |m−1 (m − 2)|y − y  |m−2

(3.43)

λ = 1, f = −g, for the interior problem, λ = −1, f = g, for the exterior one. On Lyapunov surfaces, the integral operator K *H with the kernel k*H is weakly singular, and it is compact in C(Γ). Using the same technique of the Green’s formulas and general spectral properties of a compact operator, we come to the following. Proposition 3.3. Characteristic values λ i of (3.42) are real numbers. There exist such positive a and b that, first, if H ≥ 0, then λ i are either less than or equal to −a, or greater than or equal to 1 + b. Second, if H ≤ 0, then λ i are less than or equal to −(1 + b), or greater than or equal to a. It is worth noting that an actual arrangement of λ i strongly depends on a boundary configuration and on the function H(y). For example, if Γ is a sphere and H > 0 is constant on Γ, then [111]

56 | 3 Walk on Boundary for Laplace equation

λi =

2i − 1 , 1 − 2H

i = 1, 2, . . . .

1 So we see that for H < 0.5, all λ i are positive, λ i ≥ 1−2H = 1 + b and the Neumann series 1 converges. This convergence holds for H > 1 also. In this case, λ i ≤ 1−2H = −a < −1. If 1 < H < 1, then | λ | ≤ 1 and those analytical continuation methods that have been used 1 2 in Chapter 3.2 for the Dirichlet BVP are applicable here. In a general case, it is known [87] that for sufficiently small H, |λ1 | is greater than 1, and hence the Neumann series converges. What’s more, this convergence holds even if we substitute the kernel k*H by its modulus. So we can use the standard method of a Monte Carlo estimator construction with a non-zero probability of a Markov chain termination. However, in every concrete case, we have to calculate the value of a beforehand, while it is not necessary to know the exact value of λ1 . Suppose that 1 ≥ |λ1 | ≥ a, where a can be calculated by using some iterative procedure, a Monte Carlo algorithm, in particular [23]. The spectral parameter substitution λ = λ0 2aη 1−η , where λ0 = 1 for the interior problem and λ0 = −1 for the exterior one, and corresponding rearrangement of the integral equation lead to the representation of a solution in the form of a convergent series for the integral operator

K1* = η* I + 2aη* λ0 K *H , 1 (Chapter 2.2). Both scalar and vector algorithms with a resolvent where η* = 1+2a iteration can be used here also, and a can be taken as a parameter of these methods. The substitution (Chapter 2.3)

λ=

4aη (1 − η)2

also can be used here, since images of points lying on the ray λ ≥ 1 + b have moduli, which are greater than  η* = 1 + 2a − 2 a2 + a. Suppose now that we know both limits of the characteristic set for the operator K *H . Then the following spectral parameter substitutions can be used here: λ=

4a(1 + b)η (1 + b)(1 − η)2 + a(1 + η)2

for the interior problem, and λ=

4a(1 + b)η a(1 − η)2 + (1 + b)(1 + η)2

for the exterior problem. These functions conformally map the unit disk, {η : |η| < 1}, onto the complex plane with two rays excluded: λ ≤ −a, λ ≥ 1 + b and λ ≤ −1 − b, λ ≥ a, respectively.

3.5 Third boundary value problem

|

57

Finally, all the considered transformations of the initial Neumann series lead to a representation of a solution in the form of a finite sum of integral functionals: # " n  (n) i i * * l i λ K H f , h x + δ(x; n) u(x) = i=0

=

n 

 f , λ i K Hi h*x + δ(x; n). l(n) i

i=0

To construct a Monte Carlo estimator based on this representation, we have to choose a Markov chain with a transitional density consistent with the kernel k*H , or what is equivalent consistent with the kernel k H . To start with, consider the density of the isotropic WB-process. Let Γ be convex, then k*H (y, y ) H(y)|y − y | , = 1 − p(y , y) (m − 2) cos φ yy

(3.44)

and this weight coefficient can be unbounded. For example, this can happen in the case when Γ has plane parts. Hence, we come to a conclusion that the isotropic WB-process is applicable for solving the third BVP only in the case of strictly convex boundaries. Consider a direct estimator now: ξ H (x; n) =

n 

i * l(n) i λ Q H,i h x (y i ),

(3.45)

i=0

where k* (y , y ) f (y0 ) , Q H,i = Q H,i−1 H i i−1 , p0 (y0 ) p(y i−1 , y i ) 2 1 * h x (y i ) = · (m − 2)σ m |x − y i |m−2 Q H,0 =

and {y0 , y1 , y2 , . . .} is a Markov chain of WB-process constructed in accordance with some initial p0 and transitional p densities. We see that conditions for p0 are not very strict and it is sufficient to have α2 ≥ p0 (y0 ) ≥ α1 > 0 for some constant α1 , α2 . As for the transitional density, we can proceed as follows. Let y i−1 ∈ Γ be already defined. Since k*H (y i , y i−1 ) is weakly singular as y i approaches y i−1 , then to have Q H,i bounded, we must include this singularity into the transitional density. So we divide a boundary surface into two parts: some neighbourhood C of the point y i−1 , C = {y : |y − y i−1 | < r0 } and all the remaining boundary Γ \ C. With some probability ω, a point y i is sampled in C in accordance with some singular density p, and with the complementary probability 1 − ω, we use some arbitrary density π(y) ≥ α > 0 to sample the next point in Γ \ C. The choice of a

58 | 3 Walk on Boundary for Laplace equation

singular density in C depends on the geometric properties of a surface Γ at the point y i−1 . Let m = 3 for simplicity. With probability 1, y i−1 is either an elliptic point or an interior point of a flat part of the boundary. Suppose that curvature of Γ at this point is finite. From here, we can conclude that the fraction in (3.44) is a bounded function of y in C for some r0 , y = y i−1 . It means that the isotropic transitional density can be used in C in this case, and the resulting transitional density on the whole Γ is p1 (y i−1 , y i ) = ωp(y i−1 , y i ) · I C (y i ) + (1 − ω)π(y i )I Γ\C (y i ),

(3.46)

where  I C (y) =

1, 0,

y∈C y ∈/ C

is the indicator function of the set C. Let y i−1 lie on a flat part of Γ. Introduce the local polar coordinate system in this plane setting y i−1 = (0, 0) and y i = (r, φ). Obviously, there exists such r0 that C is a circle lying in this plane. We can choose the function (2πr0 |y i − y i−1 |)−1 as a density in this circle, and so we have p1 (y i−1 , y i ) = ω(2πr0 |y i − y i−1 |)−1 · I C (y i ) + (1 − ω)π(y i )I Γ\C (y i ).

(3.47)

It means that with probability ω, y i ∈ C, and the weight coefficient is equal to k*H (y i , y i−1 ) 1 = − H(y i )r0 , ω p1 (y i−1 , y i ) since cos φ y i y i−1 = 0 here, and with probability 1 − ω, y i ∈ Γ \ C, and k*H (y i , y i−1 ) k* (y , y ) 1 = · H i i−1 . p1 (y i−1 , y i ) 1 − ω π(y i ) In Chapter 3.4, we considered the exterior Dirichlet problem and represented its solution as a sum (3.34) of two functions: a double-layer potential with a density μ and the Robin’s potential. The last one is equal to O(|x|2−m ) when |x| → ∞. This feature ensures the most general harmonic properties at infinity for the solution of BVP (3.31). The same result can be obtained if instead of the Robin’s potential we use a single-layer potential with the density equal to −Hμ, where H is some continuous function. In particular, H may be equal to some non-zero constant value on Γ. So, consider the Dirichlet problem and represent its solution in the form u(x) = W[μ](x) + V[−Hμ](x).

(3.48)

3.5 Third boundary value problem

|

59

As a consequence, we come to the integral equation μ = λK H μ + f , where K H is the integral operator adjoint to K *H . Its kernel k H (y, y ) is equal to k*H (y , y), which is defined in (3.43). For the interior problem, λ = −1 and f = g, while for the exterior one, λ = 1 and f = −g. From Statement 3.2, it follows that the spectral properties of the integral operator K H are identical to the spectral properties of K *H . Hence, the following proposition is valid. Theorem 3.8. The Dirichlet problem is solvable for all continuous boundary values g and its solution can be represented as a sum of two potentials, W[μ] and V[−Hμ], where H ≤ 0 for the interior problem and H ≥ 0 for the exterior one. Since k H (y, y ) = k*H (y , y), it means that the same WB-process that has been constructed for solving the third BVP is applicable here. The main difference is that we use the adjoint estimator here: ξ H* (x; n) =

n 

i * l(n) i λ Q H,i f (y i ),

(3.49)

i=0

where

  cos φ y0 x 1 H(y0 ) 2 , − · p0 (y0 ) σ m |y0 − x|m−1 (m − 2)|y0 − x|m−2 k (y , y ) Q*H,i = Q*H,i−1 H i−1 i , p(y i−1 , y i )

Q*H,0 =

and the transitional density, p, is either isotropic for strictly convex boundary or it is equal to p1 defined in (3.46), (3.47). Another alternative method of solving Dirichlet BVPs is based on the representation of a solution in the form of a single-layer potential. As a result, we come to the need of solving boundary integral equation of the first kind:  μ(y ) 1 · dσ(y ) = g(y), y ∈ Γ. (3.50) (m − 2)σ m |y − y |m−2 Γ

To construct a stochastic estimator for a solution of the Dirichlet problem (for exterior and interior simultaneously), we can use the results of Section 2.5. Here, we have u(x) = (h x , μ), where h x (y) =

1 |x − y |2−m , (m − 2)σ m

60 | 3 Walk on Boundary for Laplace equation

and k(y, y ) = k(y , y) =

1 |y − y  |2−m . (m − 2)σ m

Note that the transitional densities (3.46), (3.47) are consistent with this kernel. Hence, they can be used for construction of the estimator.

3.6 Inhomogeneous problems Up to this point, everywhere in Chapter 3, the Laplace equation has been considered. Suppose now that its right-hand side is non-zero, and so we come to the need of solving some BVP for the Poisson’s equation: x ∈ G,

∆u(x) = −F(x), Bu(y) = g(y),

y ∈ Γ,

where B is a boundary operator of the first (B = I), second (B = (B =

(3.51) ∂ ), ∂n(y)

or third BVP

∂ ∂n(y)

− H(y)). Consider Newton’s volume potential,  1 F(y) · dy. V0 (x) = (m − 2)σ m |x − y|m−2

(3.52)

G

Let F(y) be continuous function in a bounded domain G. Then V 0 ∈ C1 (Rm ), it is harmonic in G1 , and V0 (x) = O(|x|2−m ) for |x| → ∞. It is well known [120] that even in the case when F is finitary distribution (generalized function), the potential V0 satisfies Poisson’s equation ∆V0 = −F, ¯ then this equation is satisfied at every point x ∈ G. As a and if F ∈ C1 (G) ∩ C(G), result, we can proceed in the following way. Consider BVP (3.51) for u and represent its solution as a sum of two functions: u(x) = v(x) + V0 (x). Hence, the auxiliary function v(x) satisfies the Laplace equation, and it is a solution of the correspondent BVP ∆v(x) = 0,

x ∈ G,

Bv(y) = g(y) − BV0 (y),

y ∈ Γ.

(3.53)

It is essential to note that we have to know boundary values of the integral (3.52) or its normal derivative at every point y on the boundary. In contrast to deterministic

3.6 Inhomogeneous problems

|

61

approach, when considering BVP (3.51) in the context of Monte Carlo methods, we can easily overcome this difficulty. The double randomization principle makes it possible to use a random estimate instead of exact value of a multidimensional integral BV0 . These considerations are valid for the exterior BVP as well, x ∈ G1 ,

∆u(x) = −F(x),

y ∈ Γ.

Bu(y) = g(y),

Some additional restrictions must be imposed on the function F here. We can require, for example, that it has a compact support. Consider another approach to solving BVP (3.51). Suppose that we construct such sufficiently smooth function u2 (x) that at every point y ∈ Γ, u2 (y) = g(y). To find such function, one of the possible ways could be the R− function method (for details, see [98]). Next we consider Dirichlet BVP (3.51) for Poisson’s equation and represent its solution in the form of a sum x ∈ G.

u(x) = u1 (x) + u2 (x),

(3.54)

As a result, we come to the BVP for the Poisson’s equation with zero boundary values ∆u1 (x) = −F(x) − ∆u2 (x), u1 (y) = 0,

x ∈ G,

y ∈ Γ.

Another way of transition from BVP (3.51) to a homogeneous BVP for the Poisson’s equation is following. We can consider function u2 (x) to be a solution to the correspondent BVP for the Laplace equation x ∈ G,

∆u2 (x) = 0,

Bu2 (y) = g(y),

y ∈ Γ.

∆u1 (x) = −F(x),

x ∈ G,

(3.55)

Then, we have

Bu1 (y) = 0,

y ∈ Γ.

To solve this problem, the following algorithm can be proposed. Let u δ (x, x0 ) be the influence function, i.e., for every x0 ∈ G, ∆ x u δ (x, x0 ) = δ(x − x0 ), B y u δ (y, x0 ) = 0,

y ∈ Γ.

x ∈ G,

x  = x0 ,

62 | 3 Walk on Boundary for Laplace equation

We see that u δ is the Green’s function in the case of the Dirichlet BVP. It is obvious that function u1 can be represented in the form of an integral functional  u1 (x) = −(u δ , F) = − u δ (x, x0 )F(x0 )dx0 . (3.56) G

Represent u δ as a sum of the fundamental solution for the Laplace equation and some function u0 : u δ (x, x0 ) = −

1 |x − x 0 |2−m + u 0 (x, x 0 ). (m − 2)σ m

(3.57)

Then, we have that u0 is a solution to the BVP for the Laplace equation x ∈ G, 1 B y (|y − x0 |2−m ), B y u0 (y, x0 ) = (m − 2)σ m ∆ x u0 (x, x0 ) = 0,

y ∈ Γ.

(3.58)

So we see that a solution of the problem (3.51) is represented as a linear functional of solutions u2 and u0 to the BVPs for the Laplace equations (3.55) and (3.58), respectively. Note that the inverse transition from u2 and u0 does not present any challenge to Monte Carlo methods. Denote by ξ0 (x, x0 ) a random estimator for u0 (x, x0 ), and by ξ2 (x), we denote a random estimator for u2 (x). Randomizing the integration in (3.56), we choose a random point x0 in G with a probability density π(x0 ). Then, according to the double randomization principle, we have that   1 F(x0 ) 2−m ξ (x) = ξ2 (x) + |x − x0 | − ξ0 (x, x0 ) π(x0 ) (m − 2)σ m is an estimator for a solution of BVP (3.51) for Poisson’s equation. Properties of this estimate (unbiasedness, consistency, finiteness of variance, etc.) are obviously determined by the correspondent properties of the random estimates ξ0 , ξ2 for a solution of the Laplace equation.

3.7 Continuity BVP Consider the problem of computing a function, u, which satisfies the Poisson equation inside a compact domain, G: ϵ i ∆u(x) = −ρ(x), x ∈ G,

(3.59)

the Laplace equation outside the domain: ϵ e ∆u(x) = 0, x ∈ Rm \ G,

(3.60)

3.7 Continuity BVP |

63

and the continuity conditions on the boundary of G: u i (y) = u e (y), ϵ i

∂u i ∂u e = ϵe , y ∈ ∂G. ∂n(y) ∂n(y)

(3.61)

Here, u i and u e are the limit values of the solution from inside and outside, respectively; ϵ i and ϵ e are constants such that ϵ i < ϵ e . With the assumption that ∂G is smooth enough, it is possible to represent the solution as a sum of a volume and single-layer potentials [32, 34],  2 1 u(x) = g(x) + μ(y)dσ(y) ≡ g(x) + u0 (x), (3.62) σ m |x − y|m−2 ∂G



ρ 1 G (m−2)σ m ϵ i |x−y|m−2

dy. where g(x) = Taking into account boundary conditions (3.61) and discontinuity properties of the single-layer potential’s normal derivative [34], we arrive at the integral equation for the unknown density, μ: μ = −q0 K μ + f , which is valid almost everywhere on ∂G. Here, q0 = 2 cos ϕ yy σ m |y−y |m−1 , 

(3.63) ϵ e −ϵ i ϵ e +ϵ i , and the kernel of the integral

operator K is where ϕ yy is the angle between (note!) external normal vector n(y) and y − y . The free term of this equation equals f = q0

∂g , ∂n(y)

and it can be computed analytically. Since q0 < 1, the Neumann series for (3.63) converges (see, e.g. [32, 34]), and it is possible to calculate the solution as u0 (x) =

∞ 

(h x , (−q0 K)i f ),

(3.64)

i=0 2ϵ i 1 where h x (y) = σ2m |x−y| m−2 . Usually, however, ϵ e  ϵ i and, hence, || λ 1 | − 1| = ϵ e +ϵ  1. i Here, λ1 = −1/q0 > 1 is the smallest characteristic value of the operator −q0 K. This means that convergence in (3.64) is rather slow, even for convex G. To speed up the convergence in (3.64), we apply the method of spectral parameter substitution considered in the previous sections (see, e.g. [46, 98, 104] for Monte Carlo algorithms based on this method). This means we consider the parameterized equation

μ λ = λ(−q0 K)μ λ + f and analytically continue its solution given by the Neumann series for |λ| < |λ1 |. This goal can be achieved by substituting in λ its analytical expression in terms of another complex parameter, η, and representing μ λ as series in powers of η.

64 | 3 Walk on Boundary for Laplace equation

In this particular case, it is possible to use the substitution λ = hence u0 (x) =

n 

 i i h l(n) (−q ) , K f + O(γ n+1 ), x 0 i

2|λ1 |η 1−η

≡ χ(η), and

(3.65)

i=0

where γ =

1 1+2|λ1 |


2(1 − β). Next, consider the normal component of the derivative and denote  ∂u(x) = g(x, t, y)μ(y)dσ(y), ∂n(t) Γ

68 | 3 Walk on Boundary for Laplace equation

where g(x, t, y) =

1 cos(n(t), y − x) . 2π |y − x |2

Divide as usual Γ into two parts: Γ0 and Γ \ Γ0 , where Γ0 = {y : y ∈ Γ, |x − y| = l < l0 } for some fixed l0 > |x − t|. With some probability α0 , we choose y on Γ0 , and with probability 1 − α0 , we simulate y on Γ \ Γ0 . First, consider t lying in the plain, and suppose that Γ0 is totally contained in this plain. Denote δ = |x − t|. Then, 

2π l0 g(x, t, y)μ(y)dσ(y) = dψ 0

Γ0

δ g(x, t, l, ψ)μ(l, ψ)dl. 2πl2

δ

The sampling procedure directly follows from this representation. We choose the angle ψ uniformly in [0, 2π] and l=

l0 δ , l0 (1 − α) + δα)

where α is a random number uniformly distributed in [0, 1]. It means that the distribution density of the point y is equal to δl0 2πl2 (l

0 − δ)

.

So we have ∂u(x) = Eζ n (x) + ε(n) (x), ∂n(t)   1 δ ξ * (y), y ∈ Γ0 ζ n (x) = · 1− α0 l0 =

1 g(x, t, y) * · ξ (y), 1 − α0 p (y)

y ∈ Γ \ Γ0 ,

(3.73)

where p (y) is a density consistent with g. Consider the elliptic case now and denote by R = R(ψ) the curvature radius of the surface Γ at the point t in the plain P(ψ). This plain contains n(t), and the angle between P(ψ) and τ1 (t) is equal to ψ. Consider a surface Γ1 defined as follows: its section with P(ψ) is a circle of radius R(ψ) with its centre at the point t + n(t)R(ψ).

3.9 Normal derivative of a double-layer potential

|

69

Then, g(x, t, y)

dσ (y ) dσ(y) = g(x, t, y1 ) 1 1 (1 + O(l)), for l → 0. dldψ dl1 dψ

(3.74)

Here, y1 ∈ Γ1 , y ∈ Γ0 and l1 = |x − y1 | = l, σ1 is the surface measure on Γ1 . Expression in the right-hand side of (3.74) can be written out explicitly: l21 + δ(2R + δ) . 4πR(R + δ)2 l21 So, if we want to obtain an estimator with finite variance, it is necessary to sample the point y on Γ0 in accordance with the density p1 = B

l2 + δ(2R + δ) , 2πl2

where B is a normalizing constant. It means that angle ψ is chosen uniformly in [0, 2π], next radius R(ψ) is calculated and then l is sampled in accordance with the formula 1/2 α α 2 l = −R + . + (R − ) + δ(2R + δ) 2B 2B Finally, we have 1 * ξ (y), y ∈ Γ0 α0 p1 1 g(x, t, y) * = · ξ (y), 1 − α0 p (y)

ζ n (x) =

y ∈ Γ \ Γ0 .

(3.75)

Next we substitute (3.72), (3.73), (3.75) in (3.71) and see that the proper choice of the modelling procedures provides a possibility to construct Monte Carlo estimators for the potential derivatives even near the boundary.

3.9 Normal derivative of a double-layer potential In contrast to other sections, here we consider the 2D case. Suppose that we seek for solution of a BVP in the form of a double-layer potential with density μ:  ∂g (x, y )dσ(y ) W(x) = μ(y ) ∂n(y ) Γ     ∂g  ∂g n1 (y )  + n2 (y )  (x, y )μ(y )dσ(y ). = ∂y1 ∂y2 Γ

70 | 3 Walk on Boundary for Laplace equation

Here g(x, y) =

1 1 . ln π |x − y|

Frequently, it is desired to find not only a solution but also its derivatives, and a normal derivative, in particular. To calculate this function, we apply the technique described in [34]. Let x ∈ G and y ∈ Γ; then, we have    ∂g ∂ ∂ ∂W n1 (y )  − n2 (y )  = (x, y )μ(y )dσ(y ), (3.76) ∂x1 ∂y2 ∂y1 ∂y2 Γ

since ∂g ∂g =−  ∂x i ∂y i and ∆ x g = 0. Following [34], we introduce differential combinations Di μ = B

∂f ∂f − n (y ), ∂yi ∂n(y ) i

where f is a function defined in G, continuously differentiable in this closed domain, and satisfying boundary conditions Bf = μ. Here, we denote by Bf the boundary value of the function f . Next, we integrate (3.76) by parts and use the fact that    ∂  ∂ dσ(n1 D2 − n2 D1 )f = −dy1  − dy2  f = −df ∂y1 ∂y2 is the total differential, and therefore integral of df along a closed curve is equal to zero. So we have     ∂W ∂g ∂g ∂g    = D μ +  D2 μ dσ(y ), (x, y )D1 μdσ(y ) − n1 (y ) ∂x1 ∂y1 1 ∂y2 ∂n(y ) Γ

Γ

and as a consequence, after transition to the local coordinate system based on the normal and tangent vectors,      ∂g ∂g ∂μ ∂W   dσ(y ) n(y) · (x, y ), −  (x, y ) = lim ∂y2 ∂y1 ∂n(y) x→y ∂τ(y ) Γ

3.9 Normal derivative of a double-layer potential

|

71

since Di f = τi

∂f ∂τ

in the 2D case. Here, we use the notations n(y) = (n1 (y), n2 (y)) and τ(y) = (n 2 , −n1 ). Finally, if we use the limiting properties of a single-layer potential’s derivatives [34],  ∂W ∂μ ∂g (3.77) = · (y, y )dσ(y ). ∂n(y) ∂τ(y ) ∂τ(y) Γ

So, we have represented the normal derivative of a double-layer potential in the form of a singular integral (3.77). Monte Carlo algorithms for calculating such integrals are given in Chapter 5. 3D case is considered there; however, it is essential that the main idea of constructing an estimator based on a pair of symmetrical points is applicable in the 2D case also. Introduce a local coordinate system at the point y in such a way that y = (0, 0), y1 = (r, θ), y2 = (r, π − θ), where θ is the angle between τ and y1 − y. Next, we sample y1 according to some angular density p θ symmetrical with respect to n(y). Hence, y2 is distributed with the same angular density. The distribution density of points y i , i = 1, 2 in the measure σ is equal to p i (y, y i ) = p θ ·

cos φ y i y . |y − y i |

Therefore, we can use the following random estimator for ∂W /∂n(y): ξ [W n ](y) =

∂μ/∂τ(y1 ) ∂μ/∂τ(y2 ) + . p1 (y, y1 ) p2 (y, y2 )

(3.78)

From here, we see that the tangential derivative ∂μ/∂τ must be known on the boundary. According to the double randomization principle, a Monte Carlo estimator can be used instead of the exact value of this function. If the Dirichlet BVP is considered, we have an integral equation and, hence, a random estimator for μ only. Nevertheless, correlated WB-processes starting from the points y1 , y2 can be used here to obtain some random estimators for the tangential derivatives at these points.

4 Walk-on-boundary algorithms for the heat equation 4.1 Heat potentials and Volterra boundary integral equations Let G be a bounded domain in Rm , m ≥ 3, and let the boundary of G, ∂G, be a piecewise Lyapunov-continuous surface. We denote by Z0 (x, t) the fundamental solution to the heat equation:   m |x |2 Z0 (x, t) = Θ(t)(4πt)− 2 exp − 4t   ⎧ 2 ⎨(4πt)−m/2 exp − |x| , t ≥ 0, |x|  = 0, 4t = (4.1) ⎩ 0, t < 0, where Θ(t) is the Heaviside step function:  0, Θ(t) = 1,

t 0, ∂t u(x, 0) = 0, x ∈ G(G1 ), u(y, t) = g(y, t), y ∈ Γ, t > 0.

(4.8)

We seek for the solution to (4.8) in the form of the double-layer potential W. From (4.7) we get the boundary integral equation for the density μ: τ μ(y, τ) = λ 0

dτ



|y − y |

τ − τ

cos φ y y Z0 (y − y , τ − τ )μ(y , τ )dσ(y ) + f (y, τ), (4.9)

Γ

where λ = −1, f (y, τ) = g(y, τ) for the interior problem and λ = 1, f (y, τ) = −g(y, τ) for the exterior problem.

74 | 4 Walk on Boundary for heat equation

Analogously, for the solution of the interior (exterior) Neumann problem ∂u = ∆u(x, t), x ∈ G(G1 ), t > 0, ∂t u(x, 0) = 0, x ∈ G(G1 ), ∂u = g(y, t), y ∈ Γ, t > 0, ∂n y

(4.10)

we use the representation in the form of the single-layer potential V. Then (4.6) gives τ dτ

μ(y, τ) = λ 0





|y − y |

τ − τ

cos φ yy Z0 (y − y , τ − τ )μ(y , τ )dσ(y ) + f (y, τ), (4.11)

Γ

where λ = 1, f (y, τ) = −g(y, τ) for the interior problem and λ = −1, f (y, τ) = g(y, τ) for the exterior problem. Finally, let us consider the third BVP: ∂u = ∆u(x, t), x ∈ G(G1 ), t > 0, ∂t u(x, 0) = 0, x ∈ G(G1 ), ∂u − H(y, t)u(y, t) = g(y, t), y ∈ Γ, t > 0. ∂n y

(4.12)

If we seek for the solution to (4.12) in the form of the single-layer potential, V, then μ(y, τ) solves the equation τ dτ

μ(y, τ) = λ 0



 

|y − y |

τ − τ

 cos φ yy − H(y, τ)

Γ

×Z0 (y − y , τ − τ )μ(y , τ )dσ(y ) + f (y, τ),

(4.13)

where λ = 1, f = −g for the interior problem and λ = −1, f = g for the exterior problem. Note that integral equations (4.9), (4.11), (4.13) are of the Volterra’ type, and their kernels include weak singularities (weak polar kernels). This implies that they have solutions for arbitrary λ, and functions μ are as smooth as the functions g(y, τ) [60]. Thus, for g ∈ C(S T ), densities μ belong to C(S T ); potentials V and W solve the heat equation and satisfy the corresponding initial and boundary conditions.

4.2 Nonstationary walk-on-boundary process Let X = Γ × (−∞, ∞). First, we assume that G is convex, and let p0 (x, τ) be a probability  density in X: p0 (y, τ) ≥ 0, and p0 (y, τ)dσ(y)dτ = 1. X

4.2 Nonstationary walk-on-boundary process

|

75

Definition 4.1. We call a homogeneous Markov chain {(y0 , τ0 ), (y1 , τ1 ), . . . } defined on X the random walk-on-boundary B-process if its states are defined as follows: the point (y0 , τ0 ) is distributed in accordance with a density p0 (y, τ), and the transitional density from (y, τ) to (y , τ ) is p(y, τ → y , τ ) =

|y − y |

τ − τ

cos φ y y Z0 (y − y , τ − τ ).

(4.14)

Note that p(y, τ → y , τ ) = 0 for τ ≥ τ, so in the trajectories of B-process, time is monotonically decreasing. We will use the factorized representation: p(y, τ → y , τ ) = p y (y → y )p τ (τ → τ |y, y ),

(4.15)

where p y (y → y ) =

2 cos φ y y σ m |y − y |m−1

(4.16)

is the spatial transitional density of the stationary walk-on-boundary process introduced in Chapter 3, and 

p τ (τ → τ |y, y ) =

π m/2 |y − y |m ( ) Z (y − y , τ − τ ) ) 0 Γ m (τ − τ 2

(4.17)

is the transitional density in time; Γ(·) is the Euler Γ-function. Thus, (4.15) gives the following algorithm of simulation of the transition (y, τ) →   (y , τ ) : (a) simulate a random point y ∈ Γ using p y (y → y ), i.e., in accordance with the isotropic distribution in the solid angle of view from the point y, that is, in the same way as in the stationary walk-on-boundary process, described in Chapter 3; (b) for y, τ, y fixed, the parameter τ (‘time’) is sampled in (−∞, τ) using the density p τ (τ → τ |y, y ), i.e., τ = τ −

| y − y  |2 , 4γm/2

(4.18)

where γm/2 is γ -distributed random variable with parameter m/2. Its density is q m/2 (x) = x m/2−1 exp(−x)/Γ(m/2),

x > 0.

Remark 4.2. If m is even, then the simulation formula is γm/2 = − ln(α 1 . . . α m/2 ),

as while for odd m, γm/2 = ξ + ζ ,

where ξ = γ[ m ] and ζ = η2 /2; η is a standard Gaussian random variable. 2

76 | 4 Walk on Boundary for heat equation

Definition 4.3. We define F-process of random walk on boundary as a homogeneous Markov chain on X with the initial density p0 and the transitional density p(y, τ → y , τ ) =

|y − y |

τ − τ

cos φ y y Z0 (y − y , τ − τ).

(4.19)

It is clear that τ > τ with probability 1. This means that time is monotonically increasing. Simulation of the F-process: (a) first, a point y is sampled on Γ exactly as in the case of B-process; (b) for fixed y, τ, y , time τ is simulated by τ = τ + |y − y |/4γm/2 , which differs from (4.18) only by the sign. The following simple statement shows the relation between B- and F-processes. Proposition 4.4. Let {(y n , τ n )}∞ n=0 be B-process on Γ with the initial density p 0 (y, τ), and let t be a fixed real number. Then {(y n , t − τ n )}∞ n=0 is F-process on Γ with the initial density p1 (y, τ) = p0 (y, t − τ). And vice versa, if {(y n , τ n )}∞ n=0 is F-process on Γ with the initial density p 0 (y, τ), then {(y n , t − τ n )}∞ is B-process on Γ with the initial density p1 (y, τ) = p0 (y, t − τ). n=0 Proof. The proof follows from the definition of B- and F-processes and the well-known rules of transformation of distribution densities when transforming the random variables. Consider a non-convex domain now. In this case, the straight line A y = {y : y = y + tω; t ∈ R}, ω − isotropic intersects the boundary Γ, say, in q(y, y ) points (not counting the point y). Then, we choose one of them at random with probability 1/q. In this case, the transitional densities of B- and F-processes are p(y, τ → y , τ ) = =

2| cosy y | π m/2 |y − y |m · Z (y − y , τ − τ )   m−1 ) 0 σ m q(y, y )|y − y | Γ( m )(τ − τ 2

| cos φ y y ||y − y  |

q(y, y )(τ − τ )

Z0 (y − y , τ − τ )

(4.20)

and p(y, τ → y , τ ) = respectively.

| cos φ y y ||y − y  |

q(y, y )(τ − τ)

Z0 (y − y , τ − τ),

(4.21)

4.3 The Dirichlet problem

|

77

Remark 4.5. Sometimes, it is necessary to construct some modifications of B- and F-processes. In particular, we can introduce a random termination of the chain or change the transition law in space and time. For example, for the third BVP, we will define a modification of F-process introducing a non-isotropic transition.

4.3 The Dirichlet problem As follows from the results of Section 4.2, to find the solution to interior and exterior Dirichlet problem, it is necessary to evaluate linear functional (4.3) of solutions to Volterra’ integral equation (4.9). Since the Neumann series for this equation converge (the relevant spectral radius is equal to zero), we can use the standard Monte Carlo estimator described in Chapter 2. We will see that the kernel of integral equation (4.9) generates B-process. It is convenient to extend the domain of definition of the functions μ and f in (4.9) from Γ × (0, ∞) to X = Γ × (−∞, ∞) by putting them zero on Γ × (−∞, 0). For these functions on X, we will still use the same notations, μ and f . Then, we can rewrite (4.9) in the form  (4.22) μ(y, τ) = k(y, τ; y , τ )μ(y , τ )dσ(y )dτ + f (y, τ), X

where k(y, τ; y , τ ) is defined by k(y, τ; y , τ ) = λ

|y − y |

τ − τ

cos φ y y Z0 (y − y , τ − τ ).

(4.23)

The solution to (4.8) is the linear integral functional of μ, the solution of (4.22):  (4.24) u(x, t) = μ(y, τ)h(1) xt (y, τ)dσ(y)dτ, X

where h(1) xt (y, τ) = Θ(τ)Z 0 (x − y, t − τ) f (y, τ) = −λg(y, τ)Θ(τ),

|x − y|

t−τ

cos φ yx ,

(4.25) (4.26)

λ = −1 for the interior problem and λ = 1 for the exterior problem. First, let us suppose that G is convex. Then, the kernel k in (4.23) coincides (up to a sign) with the transitional density of B-process (see formula (4.14)). Therefore, for calculating functional (4.24), it is natural to use here B-process and the adjoint estimator (Section 2.1). In this case, it is necessary to choose the initial density of

78 | 4 Walk on Boundary for heat equation

B-process so that p0 (y, τ)  = 0 for 0 < τ < t,

(4.27)

i.e., where h(1) xt (y, τ)  = 0. Proposition 4.6. The estimator (1) = −Q0 ξ xt

∞ 

λ n+1 g(y n , τ n )Θ(τ n )

(4.28)

n=0

is unbiased, i.e., (1) , u(x) = Eξ xt

(4.29)

where {(y n , τ n )}∞ n=0 is B-process with an initial density p 0 satisfying condition (4.27) and Q0 =

h(1) Θ(τ0 )Z0 (x − y0 , t − τ0 )|x − y0 | cos φ y0 x xt (y 0 , τ 0 ) = . p0 (y0 , τ0 ) p0 (y0 , τ0 )(t − τ0 )

(4.30)

Proof. The proof follows from the fact that the Neumann series for (4.22) converges and standard arguments given in Section 2.1 Note that for the interior problem, it is possible to simplify estimator (4.28) by choosing the initial density as p0 (y, τ) = p0 (y, τ|x, t) = 

cos φ yx = σ m |y − x|m−1

|x − y | cos φ yx



2(t − τ)

Z 0 (x − y, t − τ)

 π m/2 |y − x|m Z0 (x − y, t − τ) . Γ(m/2)(t − τ)

(4.31)

To sample the first point according to this density, first simulate a point y0 ∈ Γ in the same way as in the stationary process (Chapter 3), i.e., according to the isotropic distribution in the angle of view from the point x ∈ G; then, τ0 is found from τ0 = t −

| x − y 0 |2 . 4γm/2

Let N t = max{n : τ n > 0}. Since g(y n , τ n ) = 0 if τ n < 0, λ = −1 and as a consequence of (4.30), (4.31), we thus get for the interior problem: (1) ξ xt = 2Θ(τ0 )

Nt 

(−1)n g(y n , t n ).

(4.32)

n=0

Remark 4.7. Note that in the case of the exterior problem, the initial density p0 (y, τ|x0 , t) can also be used, where x0 ∈ G is some auxiliary point. Note also that if

4.3 The Dirichlet problem

|

79

it is necessary to find u(x, t) for a small value of t, then it may happen that the use of (4.31) generates the initial time instances, most of which are negative. Therefore, it is better to take p0 (y, τ) =

cos φ yx p t (τ), σ m |y − x|m−1

(4.33)

where p t (τ) is a suitable density on (0, t). Consider now non-convex G. In this case, we use B-process {(y n , τ n )}∞ n=0 with an initial density, which is consistent with the function h(1) (y, τ), while the transitional xt density is taken from (4.20). The unbiased estimator for u(x, t) is (1) ξ xt =−

∞ 

Q n λ n+1 g(y n , τ n )Θ(τ n ),

(4.34)

n=0

where λ = −1 for the interior problem and λ = 1 for the exterior problem, and h(1) Θ(τ0 )Z0 (x − y, t − τ0 )|x − y0 | cos φ y0 x xt (y 0 , τ 0 ) = , p0 (y0 , τ0 ) (t − τ0 )p0 (y0 , τ0 ) Q n = Q n−1 · sign{cos φ y n y n−1 }q(y n−1 , y n ), n = 1, 2, . . . . Q0 =

(4.35) (4.36)

Again, since g(y n , τ n ) = 0 for τ n < 0, we get (1) ξ xt =−

Nt 

Q n λ n+1 g(y n , τ n ),

(4.37)

n=0

where N t = max{n : τ n > 0}. Remark 4.8. To find the solution at several points (x i , t i ), i = 1, 2, . . . , k, we do not need to construct k Markov chains. In fact, it is sufficient to use only one B- (or F-) process. We choose the initial density p0 (y, τ) consistent with every function h x i ,t i (y, τ), i = 1, . . . , k, i.e., p0 (y, τ)  = 0 for 0 < τ < max t i , and cos φ yx i  = 0. i

Then, = Q(i) ξ x(1) 0 i ti

Nt 

Qn λ n+1 g(y n , τ n ),

i = 1, 2, . . . , k

(4.38)

n=0

are unbiased estimators for u(x i , t i ), i = 1, 2, . . . , k. Here, Q(i) 0 =

h(1) Θ(τ0 )Z0 (x i − y0 , t i − τ0 )|x i − y0 | cos φ y0 x i x i t i (y 0 , τ 0 ) = , p0 (y0 , τ0 ) (t i − τ0 )p0 (y0 , τ0 )

Q0 = 1, Qn = Qn−1 sign{cos φ y n y n−1 }q(y n−1 , y n ),

n = 1, 2, . . . .

(4.39)

80 | 4 Walk on Boundary for heat equation β

Note also that it is easy to write down estimators for the derivatives D αx D t u(x, t). These estimators have the same form as (4.37). The difference is in the initial weight: Q0 =

D αx D βt h(1) xt (y 0 , τ 0 ) . p0 (y0 , τ0 )

4.4 The Neumann problem In this section, for functions of (y, τ) defined on Γ × [0, T], we use the same extensions as in Section 4.3. Such approach makes possible rewriting (4.11) in the following form:  μ(y, τ) = k(y, τ; y , τ )μ(y , τ )dσ(y )dτ + f (y, τ). (4.40) X

Then the solution to (4.10) is written as the linear functional of μ  u(x, t) = μ(y, τ)h(2) xt (y, τ)dσ(y)dτ,

(4.41)

X

where h(2) xt = 2Θ(τ)Z 0 (x − y, t − τ), k(y, τ; y , τ ) = λ

(4.42)



|y − y |

cos φ yy Z0 (y − y , τ − τ ), τ − τ f (y, τ) = −λΘ(τ)g(y, τ),

(4.43) (4.44)

where λ = 1 for the interior problem and λ = −1 for the exterior Neumann problem. Note that, compared to (4.23), there is an inversion y ↔ y in the argument of the cos-function. Therefore, it is convenient to choose the direct Monte Carlo estimator and use F-process. Consequently, the initial density p0 (y, τ) has to be chosen consistent with f (y, τ), i.e., p0 (y, τ)  = 0 when τ > 0 and g(y, τ)  = 0. Proposition 4.9. The random variable (2) =− ξ xt

∞ 

λ n+1 Q n h(2) xt (y n , τ n )

n=0 ∞ 

= −2

n=0

λ n+1 Q n Θ(τ n )Z0 (x−y n , t−τ n ),

(4.45)

4.5 Third BVP

|

81

where Q0 = Θ(τ0 )g(y0 , τ0 )/p0 (y0 , τ0 ), Q n = Q n−1 sign{cos φ y n y n−1 }q(y n , y n−1 ),

n = 1, 2, . . . ,

(4.46)

is an unbiased estimator for the solution to the Neumann problem. Proof. The proof follows by standard arguments, if we take into account that the transitional density of F-process {(y n , τ n )}∞ n=0 has form (4.21). Note that since Z0 (x − y n , t − τ n ) = 0 for τ n > τ, we can rewrite (4.45) in the following form: (2) ξ xt = −2

Nt 

λ n+1 Q n Θ(τ n )Z0 (x − y n , t − τ n ),

(4.47)

n=0

where N t = max{n : τ n < t}. Remark 4.10. To find the solution at several points (x i , t i ), i = 1, 2, . . . , k, we can use only one F-process, as described in Remark 4.8. The correlated estimators for u(x i , t i ) have the form ξ x(2) = −2 i ti

∞ 

λ n+1 Q n Θ(τ n )Z0 (x i − y n , t i − τ n ),

(4.48)

n=0

where the random weights Q n are defined by (4.46), and the summation in (4.48) is taken over n : τ n < t i . β For x ∈ G, the unbiased estimator for the derivative D αx D t u(x, t) is (2) ξ¯ xt = −2

Nt 

β

λ n+1 Q n Θ(τ n )D αx D t Z0 (x − y n , t − τ n ),

n=0

where λ = 1 for the interior problem and λ = −1 for the exterior Neumann problem.

4.5 Third BVP We proceed in the same way as in Sections 4.3 and 4.4 and rewrite boundary integral equation (4.13) in the form:  μ(y, τ) = k(y, τ; y , τ )μ(y , τ )dσ(y )dτ + f (y, τ). (4.49) X

82 | 4 Walk on Boundary for heat equation

Then the solution to the third BVP is given by  u(x, t) = μ(y, τ)h(3) xt (y, τ)dσ(y)dτ.

(4.50)

X

Here, (2) h(3) xt (y, τ) = h xt (y, τ) = 2Θ(τ)Z 0 (x − y, t − τ),   |y − y |  k(y, τ; y , τ ) = λ cos φ − H(y, τ) Z0 (y − y , τ − τ ), yy τ − τ

f (y, τ) = −λΘ(τ)g(y, τ),

(4.51) (4.52)

with λ = 1 for the interior problem and λ = −1 for the exterior problem. We use F-process {(y n , τ n )}∞ n=0 with an initial density p 0 (y, τ) consistent with f (y, τ). Let (3) ξ xt =−

Nt 

2λ n+1 Q n Θ(τ n )Z0 (x − y n , t − τ n ),

(4.53)

n=0

where N t = max{n : τ n < t}, Q0 = Θ(τ0 )g(y0 , τ0 )/p0 (y0 , τ0 ),   H(y n , τ n )(τ n − τ n−1 ) . Q n = Q n−1 1 − |y n − y n−1 | cos φ y n y n−1 In the next section, we will prove that the Neumann series for the integral equation (3) with the kernel |k| converges. Here, k is given by (4.51). Consequently, ξ xt is an unbiased estimator for functional (4.50). The variance of this estimator can be finite or infinite. For example, if the curvature of the boundary is arbitrary small at a point, then the variance is infinite. In this case, it is possible to construct a modified walk-on-boundary process. Indeed, let us describe such a modification. First, consider the 3D case. Let x0 be a point in G such that inf cos φ yx0 > 0.

y∈Γ

Let ω0 = (y − x0 )/|y − x0 |, and let p(ω|ω0 ) be a density (on a unit sphere) with a singularity of the type |ω − ω0 |−β (0 < β < 1) at the direction ω0 . We can construct the random distribution as follows. Choose the coordinate system in such a way that ω0 coincides with the x3 axis. Then, if ω is isotropic, the latitude angle φ is uniformly distributed on [0, 2π], and the altitude angle ψ is distributed with the density sin ψ on [0, π]. This density behaves like |ω − ω0 | at the direction ω0 .

4.5 Third BVP

|

83

If we choose ψ with the distribution density c β ψ−β on [0, π], where ⎡ cβ = ⎣

π

⎤−1 t−β dt⎦ ,

0

then the density p(ω|ω0 ) has a singularity |ω − ω0 |−β . Thus, the density cos φ yx0 | y − x 0 |2

p(ω|ω0 )

has the singularity |y − y |−β , i.e., C1 | y − y  |β

≤

cos φ yx0 | y − x 0 |2

p(ω|ω0 ) ≤

C2 | y − y  |β

,

C1 , C2 > 0.

Consider the distribution density of the transition (y , τ ) → (y, τ): p3 (y , τ → y, τ) =

cos φ yx0 · p(ω|ω0 ) | y − x 0 |2   | y − y  |2   |y − y | × √ . exp − 4(τ − τ ) π(4(τ − τ ))1/2 (τ − τ )

(4.54)

Thus, to simulate (y , τ ) → (y, τ), we first sample the direction ω in accordance with p(ω|ω0 ) and take the next point in space, y, as the point of intersection of a ray, starting from x0 and having the direction ω, with the boundary Γ. In this case, y is distributed with the density cos φ yx0 · p(ω|ω0 ) . | y − x 0 |2 The next instant of time is simulated using the formula τ = τ + |y − y |2 /4γ1/2 , where γ1/2 is a sample value of the γ -distributed random variable with the parameter equal to 1/2. This implies that τ is distributed on (τ , ∞) with the following density:   |y−y |2 |y − y  | exp − 4(τ−τ ) . p(τ|τ , y, y ) = √ π(τ − τ )[4(τ − τ )]1/2 In two dimensions, we can use the density p2 (y , τ → y, τ) =

cos φ yx0 · p(ω|ω0 ) |y − x0 |   |y − y  |β exp{−|y − y  |2 /4(τ − τ )} , × Γ(β/2)(τ − τ )(4(τ − τ ))β/2

(4.55)

84 | 4 Walk on Boundary for heat equation where p(ω|ω0 ) is a density on the unit circle, having a singularity of the type |ω−ω0 |−β . The spatial sampling is the same as in the 3D case, and the next instant of time is taken as τ = τ + |y − y |2 /4γ β , 2

where γβ/2 is a sample value of the γ -distributed random variable with the parameter equal to β/2. Let {(y¯ n , τ¯ n )}∞ n=0 be a homogeneous Markov chain in a phase space X = Γ ×(−∞, ∞) with an initial density p0 (y, τ) consistent with function (4.52) and with transitional density (4.54) (3D) or (4.55) (2D). Then, the random variable (3) ξ¯ xt =−

Nt 

2Q¯ n Θ(τ¯ n )Z0 (x − y¯ n , t − τ¯ n )

(4.56)

n=0

is an unbiased estimator for u(x, t), the solution to the third BVP. In (4.56), Θ(τ¯ 0 )g(y¯ 0 , τ¯ 0 ) , Q¯ 0 = p0 (y¯ 0 , τ¯ 0 )   ¯Q n = Q¯ n−1 |y¯ n − y¯ n−1 | cos φ y¯ n y¯ n−1 − H(y¯ n , τ¯ n ) τ¯ n − τ¯ n−1 ×

|y¯ n − x 0 |2 4π|y¯ n − y¯ n−1 | cos φ y¯ n x0 p(ω n |ω*n−1 )

in 3D case, and   |y¯ n − y¯ n−1 | cos φ y¯ n y¯ n−1 − H(y¯ n , τ¯ n ) Q¯ n = Q¯ n−1 τ¯ n − τ¯ n−1 ×

|y¯ n − x 0 |Γ(β/2)(4π(τ¯ n − τ¯ n−1 ))β/2 4π|y¯ n − y¯ n−1 | cos φ y¯ n x0 p(ω n |ω*n−1 )

in two dimensions, where ω*n−1 = (y¯ n−1 − x0 )/|y¯ n−1 − x0 |. We analyse unbiasedness and variance of the estimator (4.56) in the next section. Remark 4.11. It is possible to weaken the condition inf cos φ yx0 > 0.

y∈Γ

Suppose that there exist points x0 , x1 , . . . , x l ∈ G such that inf

y∈Γ

l  i=0

| cos φ yx i | > 0.

4.6 Unbiasedness and variance |

85

To construct a Markov chain in this case, we can use a mixture (a linear combination) of transitional densities of type (4.54) in 3D or (4.55) in 2D.

4.6 Unbiasedness and variance of the walk-on-boundary algorithms In this section, we give a rigorous proof that all the estimators constructed in Chapter 4 are unbiased and have a finite variance. Also we estimate the cost of the walk-on-boundary algorithms in application to solving BVPs for the heat equation. All the Monte Carlo estimators constructed in this chapter are based on a reformulation of the original problem in the form of a Volterra–Fredholm integral equation of the second kind: τ dτ

μ(y, τ) = 0





k(y, τ; y , τ )μ(y , τ )dσ(y ) + f (y, τ).

(4.57)

∂G

For arbitrary (y, τ) ∈ Γ t ≡ ∂G × (0, t) and (y , τ ) ∈ Γ t , the kernels k(y, τ; y , τ ) of these equations satisfy the conditions   γ|y − y  |2   −(m/2+α)  β . (4.58) |k(y, τ; y , τ )| ≤ c[τ − τ] |y − y | exp − τ − τ Generally, positive constants c and γ in (4.58) may depend on t, and α, β are some parameters. Denote by K αβ the class of kernels satisfying this inequality. To investigate the variance and the cost of algorithms presented in this chapter, we first derive some properties of solutions to equations of type (4.58) whose kernels belong to K αβ . Lemma 4.12. If k ∈ K αβ , then k ∈ K α−ε/2,β−ε for any ε > 0. Proof. Indeed, 

 1/2

|y − y |/(τ − τ )

!ε

  δ | y − y  |2 0. Consequently, from k ∈ K αβ , we get   γ|y − y  |2 |k(y, τ; y  , τ )| ≤ c[τ − τ ]−(m/2+α) |y − y  |β exp − τ − τ    2 γ|y − y |  −(m/2+α−ε/2)  β−ε . ≤ c1 [τ − τ ] |y − y | exp − 2(τ − τ ) This property implies that it is possible to simplify the two-parameter expressions depending on α and β by passing to a one-parameter formulation.

86 | 4 Walk on Boundary for heat equation Lemma 4.13. If k ∈ K αβ where δ=

1 (β − 2α + 1) > 0, 2

β + m − 1 > 0,

(4.59)

and the free term of (4.57) satisfies the condition |f (y, τ)| ≤ c f τ a ,

(y, τ) ∈ Γ t ,

a > −1,

(4.60)

where c f does not depend on y and τ, and ∂G is a Lyapunov surface, then the Neumann series for (4.57) converges and |[K j f ](y, τ)| ≤ cc f Γ(a + 1)τ a

(Aτ δ )j , Γ(a + 1 + jδ)

(4.61)

where A is a constant, which does not depend on α, β and ∂G. Proof. We prove this property by using some general results presented in [89]. Denote by ω y (r) the surface area of a part of ∂G that lies inside a ball B(x, r):  ω y (r) = dσ(y ). |y−y | 1; γ = maxi Γ(δ i ). This completes the proof. We now turn to the estimators η(i) xt , i = 1, 2, 3, 4, and prove that they are unbiased and have finite variances. Unbiasedness of these estimators follows from the convergence of Neumann series for (4.9), (4.11), (4.13). Indeed, | cos φ| < c|y − y |λ , where φ equals to φ yy or to φ yy , 0 < λ ≤ 1, since ∂G is a Lyapunov surface. Hence, the kernels of (4.9), (4.11) belong to K1,1+λ (weakly polar kernels), and from Lemma 4.2 we conclude that the Neumann series converges if the right-hand side is bounded. Next, the kernel of (4.13) is represented as a sum of two kernels: the first kernel belongs to K1,1+λ and the second one lies in K0,0 . Thus, by Lemma 4.3, the Neumann series for (4.13) converges. To prove that the variance of η(1) xt is finite, it is sufficient to show that the Neumann series for the equation τ φ* (y, τ) = 0

dτ



2





k (y, τ; y , τ ) *   φ (y , τ )dσ(y ) + p(y, τ → y , τ )

2

h(1) xt (y, τ) p0 (y, τ)

∂G

converges. Here, k(y, τ; y , τ ) is the kernel of (4.9) and p is the transitional density of B-process. Convergence of the Neumann series follows now from the fact that k2 /p ≤ const |k|, and so k2 /p ∈ K1,1+λ . Clearly, p0 must be consistent with h(1) xt . For the estimator η(2) , the proof is fully analogous. xt

4.6 Unbiasedness and variance |

89

To prove that the variance of η(3) xt is finite, it is sufficient to prove that the Neumann series for the equation τ *

φ (y, τ) =

dτ

0



f 2 (y, τ) k2 (y, τ; y , τ ) *   φ (y , τ )dσ(y ) +   p3 (y , τ → y, τ) p0 (y, τ)

∂G

converges. Here, k(y, τ; y , τ ) is the kernel of (4.13). We have k = k1 + k2 + k3 , where |y − y |

cos φ yy Z0 (y − y , τ − τ ), τ − τ k2 = −4H(y, τ)Z0 (y − y , τ − τ ), k1 =

k3 =

4H 2 (y, τ)(τ − τ ) Z0 (y − y , τ − τ ). |y − y  | cos φ yy

Note that k1 and k2 are weakly polar kernels, while k3 becomes weakly polar if additional assumptions are made. Indeed, if cos φ yy ∼ |y − y |, then k3 ∈ K−1,−2 , and hence the condition (4.59) is not satisfied. So, we assume that the boundary ∂G satisfies the following condition: cos φ yy < c|y − y |λ1 ,

y, y ∈ ∂G,

1 > λ1 > 0.

Then, k3 ∈ K−1,−1−λ1 , and therefore the kernel k3 is weakly polar. Thus, for such boundaries, the variance of η(3) xt is finite. Now we show that the variance of η(4) xt is finite if the Lyapunov parameter λ is larger than 1/2. It suffices to prove that k2 /p4 is a weakly polar kernel, where k is the kernel of (4.13). Here, we have  | y − y  |2  c1 [τ − τ ]−3/2 |y − y |1−β exp − ≤ p4 (y , τ → y, τ) 4(τ − τ )  | y − y  |2  , ≤ c2 [τ − τ ]−3/2 |y − y |1−β exp − 4(τ − τ ) and therefore k2 /p4 = k1 (y , τ ; y, τ) + k2 (y , τ ; y, τ) + k3 (y , τ ; y, τ), where k1 ∈ K2,1+2λ+β , k2 ∈ K1,1+β , k3 ∈ K0,β−1 . Consequently, the kernels k1 , k2 and k3 are all weakly polar if 2λ + β > 2,

λ + β > 1.

(4.65)

90 | 4 Walk on Boundary for heat equation

The second inequality in (4.65) can be satisfied for arbitrary λ > 0 by an appropriate choice of β ∈ (0, 1), while the first inequality is satisfied for λ > 1/2.

4.7 The cost of the walk-on-boundary algorithms First, we consider the walk-on-boundary algorithms for solving the Neumann problem. For this problem, the solution is based on simulation of F-process, {(y n , τ n )}∞ n=0 . The estimator cost is proportional to the average number of transitions in the Markov chain. Thus, we need to estimate EN t . Proposition 4.16. Suppose that the boundary Γ satisfies the following condition (interior cone condition): there exist such two numbers α 0 ≥ 0 and h0 > 0 that for an arbitrary y ∈ Γ, the cone K y (α0 , h0 ) lies inside G. Here, the cone is defined by its vertex, y, angle, α0 , and height, h0 , and both angle and height do not depend on y. Suppose that the initial density of F-process is taken so that τ0 > 0 with probability 1. Then, the following asymptotic inequality is true: EN t ≤ ct for large t. Constant value, c, depends on α0 and h0 . Proof. Let (i) . ∆τ i = |y i − y i−1 |2 /4γm/2

Then, N t = max{n : τ0 + ∆τ1 + · · · + ∆τ n < t}. Denote ρ i = |y i − y i−1 |, and let  h0 , if the vector y i − y i−1 belongs to K y i−1 (α0 , h0 )  . ρi = 0, otherwise Then, ρ i ≥ ρi ; ρi are independent, and ρi = h0 with probability 2α0 /σ m , while ρi = 0 with probability 1 − 2α0 /σ m . (i) Let δ be a fixed positive number. We introduce a δ-cut for the sequence γm/2 by the following definition:  (i) (i) γm/2 , γm/2 ≥δ  γi = . (i) δ, γm/2 < δ

4.7 The cost of the walk-on-boundary algorithms |

91

Let ∆τi = (ρi )2 /4γi ,

i = 1, 2, . . . .

Since ∆τ i ≥ ∆τi , and τ0 is positive with probability 1, we conclude that the inequality N t ≤ N t = max{n : ∆τ1 + ∆τ2 + · · · + ∆τn < t} holds with probability 1. The random variables ∆τ1 , ∆τ2 , . . . are independent, equally distributed with a finite expectation μ and variance σ2 . Consequently, by the renewal theorem [13], the following asymptotic relation is true: EN t =

t σ2 t + O(t), DN t = 3 + O(t) μ μ

(4.66)

as t → ∞. Therefore, from N t ≥ N t , it follows that EN t ≤ C1 t,

DN t ≤ C2 t2

with probability 1. Consider now a B-process {(y n , τ n )}∞ n=0 with the probability density of the initial point, p0 (y, τ). We suppose that τ0 ≤ t with probability 1 (this is consistent with the condition p0 (y, τ)  = 0, 0 < τ < t). Let N t = max{n : τ n > 0}. Therefore, if the interior cone condition is satisfied, then (4.66) is true. The proof is the same as we presented for the F-process. (i) The estimations we obtained can be used for estimating the variance of ξ xt . (1) Indeed, let us consider, for example, the estimator ξ xt . Denote by d(x) the distance from x to the boundary Γ:

d(x) = inf {|x − y| : y ∈ Γ }. Then, if p0 (y, τ) > C3 > 0 for all (y, τ), for which h(1) xt (y, τ)  = 0, we get the following estimate:    h(1) (y , τ )  C4  xt 0 0  |Q0 | =  . (4.67) ≤  p0 (y0 , τ0 )  (d(x))m+1 (1) If the function g(y, τ) is bounded, then using the representation for ξ xt , we get from (4.66) and (4.67), (1) 2 ) ≤ E(ξ xt

C5 t2 . (d(x))2(m+1)

92 | 4 Walk on Boundary for heat equation (2) The same result can be proved for the estimator ξ xt in the case of bounded g(y, τ) and p0 > C6 > 0. Thus, the cost of the walk-on-boundary algorithms behaves asymptotically like Ct3 as t → ∞.

4.8 Inhomogeneous heat equation Consider the inhomogeneous heat equation, ∂u = ∆u(x, t) + F(x, t), x ∈ G(G1 ), 0 < t < T, ∂t

(4.68)

subject to initial condition u(x, 0) = φ(x),

x ∈ G(G1 ),

(4.69)

and boundary condition Bu|Γ = g(y, t),

(4.70)

where B is the boundary operator of the first, second or third kind. We suppose that the function F is Hölder continuous with respect to the arguments x, t and that φ is continuous. For unbounded G(G1 ), we assume in addition that the functions F, φ increase not faster than exp(a|x|2 ), as |x| → ∞; here, a is a positive constant, which depends on T. Let u0 (x, t) = J1 (x, t) + J2 (x, t),

(4.71)

where t J1 (x, t) =

dt

0



J2 (x, t) =



F(x , t )Z0 (x − x , t − t )dx ,

G(G1 )

φ(x )Z0 (x − x , t)dx .

G(G1 )

It is not difficult to verify that u0 satisfies equation (4.68) and initial condition (4.69). Consequently, the function u1 = u − u0 solves the following problem: ∂u1 = ∆u1 , x ∈ G(G1 ), 0 < t < T, ∂t u1 (x, 0) = 0, Bu1 |Γ = g1 (y, t),

y ∈ Γ,

(4.72)

4.8 Inhomogeneous heat equation

|

93

where g1 (y, t) = g(y, t) − Bu0 (y, t). Thus, if the functions u0 and Bu0 |Γ are known, a solution to inhomogeneous problem ¯ (4.68), (4.70) can be found by solving homogeneous problem (4.72). Denote by F(x, t) ¯ and φ(x) the continuations of F(x, t) and φ(x), respectively,  ¯F(x, t) = F(x, t), x ∈ G , (4.73) 0, x ∈ G  φ(x), x ∈ G ¯ φ(x, t) = . (4.74) 0, x ∈ G Hence, t J1 (x, t) =

dt

J2 (x, t) =

¯  , t )Z0 (x − x , t − t )dx , F(x

Rm

0





¯  )Z0 (x − x , t − t )dx . φ(x

(4.75)

Rm

It is clear that J1 (x, t) = Eη(0) 1 (x, t), where 1/2 ¯ ω, t − θ). η(0) 1 (x, t) = t F(x + 2[(t − θ)γm/2 ]

(4.76)

Here θ is a random variable uniformly distributed on [0, t], and ω is a unit isotropic vector in Rm . Analogously, ∂J1 (x, t) = Eη(i) 1 (x, t), ∂x i where  η(i) 1

√

=− t

2Γ

m+1 2

m Γ 2



"



θ2 F x+2 t− t

# 1/2 . γm+1 θ2 · ω, ω i , i = 1, . . . , m, (4.77) 2 t

94 | 4 Walk on Boundary for heat equation

and J2 (x, t) = Eη(0) 2 (x, t), ∂J2 (x, t) (x, t) = Eη(i) 2 (x, t), i = 1, 2, . . . , m, ∂x i where  ¯ + 2 tγm/2 ω), η(0) 2 (x, t) = φ(x

m +1 Γ / 1 2 ¯ + 2 tγ m+1 ω)ω i , i = 1, 2, . . . , m. η(i) φ(x 2 = −√ 2 t Γ(m/2)

4.9 Calculation of derivatives on the boundary We deal here with the problem of calculating the normal derivative solution of the following BVP:

∂u ∂n (y)

for the

∂u (x, t) = ∆u(x, t) + f (x, t), x ∈ G, t > 0, ∂t u(x, 0) = φ(x), x ∈ G, y ∈ Γ.

u(y, t) = 0,

(4.78)

Note that the straightforward differentiation of the double-layer potential representation, W, leads to the divergent integral as x → y ∈ Γ. To overcome this difficulty, we can use the specific feature of problem (4.78), namely, the condition u(y, t) = 0, y ∈ Γ. Denote by G0 (x, t; x ) the Green’s function, which is defined as the solution to the following BVP: ∂G0 (x, t; x ) − ∆ x G0 (x, t; x ) = δ(x − x , t), x ∈ G, t > 0, ∂t G0 (x, 0; x ) = 0, x ∈ G, x  = x , G0 (y, t; x ) = 0,

y ∈ Γ,

t > 0.

Then, we have t u(x, t) = 0

dt



+ G



f (x , t )G0 (x, t − t ; x )dx

G

φ(x )G0 (x, t; x )dx .

(4.79)

4.9 Calculation of derivatives on the boundary | 95

Let G0 (x, t; x ) = Z0 (x − x , t) + g(x, t; x ), where g is a continuous in x ∈ G function. Then, we can rewrite (4.79) in the following form: t u(x, t) =

dt

0



g(x, t − t ; x )f (x , t )dx

G





+





t

φ(x )g(x, t; x )dx + 0

G





+



dt



Z0 (x − x , t − t )f (x , t )dx

G



φ(x )Z0 (x − x , t)dx ,

(4.80)

G

where the function g has to satisfy the BVP: ∂g (x, t; x ) = ∆ x g(x, t; x ), x ∈ G, t > 0, ∂t g(x, 0; x ) = 0, x ∈ G, g(y, t; x ) = −Z0 (y − x , t), y ∈ Γ, t > 0.

(4.81)

We continue g(x, t; x ) to the whole space Rm by setting g(x, t; x ) = −Z0 (x − x , t),

(x, t) ∈ G1 × (0, ∞).

Since x ∈ G1 , the function g satisfies in G1 × (0, ∞) the following BVP: ∂g (x, t; x ) = ∆ x g(x, t; x ), x ∈ G1 , t > 0, ∂t g(x, 0; x ) = 0, x ∈ G1 , ∂g ∂Z0 (y, t; x ) = − (y − x , t) = ψ(y, t; x ), y ∈ Γ, t > 0. ∂n(y) ∂n(y)

(4.82)

Consequently, the function g can be represented in G1 × (0, ∞) as a single-layer potential t



g(x, t; x ) = 2

 dτ

0

Z0 (x − y, t − τ)μ(y, τ; x )dσ(y),

(4.83)

Γ

where the density μ solves in Γ × (0, ∞) the integral equation 

τ

μ(y, τ; x ) = − 0

dτ



|y − y | cos φ yy Z0 (y − y , τ − τ )μ(y , τ ; x )dσ(y ) |τ − τ |

Γ

+ ψ(y, τ; x ).

(4.84)

96 | 4 Walk on Boundary for heat equation The representation for g(x, t; x ) is true in G1 × (0, ∞) and for (x, t) ∈ G × (0, ∞) as well, since potential (4.83) satisfies the heat equation and the boundary condition in (4.82). Hence, we get ⎡ ⎤  τ  t−t  ⎢ ⎥ u(x, t) = dt f (x , t )⎣2 dτ Z0 (x − y, t − t − τ)μ(y, τ; x )dσ(y)⎦dx 0

G



⎡ 

φ(x )⎣2

+

0

t



⎤ 

Z0 (x − y, t − τ)μ(y, τ; x )dσ(y)⎦dx

dτ 0

G

Γ

Γ

+ J1 (x, t) + J2 (x, t).

(4.85)

Here t J1 (x, t) =

dt

0

 J2 (x, t) =



Z0 (x − x , t − t )f (x , t )dx ,

G

φ(x )Z0 (x − x , t)dx .

G

Calculate now the normal derivative of the solution to (4.78) by differentiating at x ∈ G in (4.79) and taking the limit as x → y ∈ Γ:  ∂u ∂u (x, t)n i (y) (y, t) = lim ∂x i x→y ∂n(y) m

i=1

t dt

= 0

⎡





f (x , t )

G

⎢ ×⎣μ(y, t − t ; x ) +  +

⎡

t−t



dτ 0

φ(x )⎣μ(y, t; x ) +

G

∂J1 ∂J2 + + . ∂n(y) ∂n(y)

⎤

 2

∂Z0 ⎥ (y − y , t − t − τ)μ(y , τ; x )dσ(y )⎦dx ∂n(y)

Γ

t

 dτ

0

⎤ ∂Z0 2 (y − y , t − τ)μ(y , τ; x )dσ(y )⎦dx ∂n(y)

Γ

(4.86)

Here, we have used the limit properties of the normal derivative of the single-layer potential (Section 4.1). Thus, the normal derivative is represented as a linear functional of the solution to (4.84). Let us suppose for simplicity that the domain is strongly convex (i.e., the curvature is positive and separated from zero) and φ(x) ≡ 0, x ∈ G. Monte Carlo

4.9 Calculation of derivatives on the boundary | 97

estimators for t

dt

0



f (x , t )μ(y, t − t ; x )dx ,

∂J1 ∂n

and

∂J2 ∂n

G

are already constructed in Section 4.2. Therefore, we need to describe the algorithm for calculating ⎡  ⎤ t  t−t  ∂Z0 ⎢ ⎥ (y − y , t − t − τ) · μ(y , τ; x )dσ(y )⎦dx . J3 (y, t) = dt f (x , t )⎣ dτ 2 ∂n(y) 0

0

G

Γ

Let p(x , t ) be a density in G × (0, t) consistent with f (x , t ). Then, we can use the following random estimator: N t−t

η yt =



(−1)n Q n ψ(y n , τ n ; x ),

(4.87)

n=0

where {(y n , τ n )}∞ n=0 is F-process with the initial density ∂Z0 (y0 − y, t − t − τ0 ) ∂n(y0 ) |y − y0 | = cos φ y0 y Z0 (y0 − y, t − t − τ0 ), |t − t − τ0 |

p0 (y0 , τ0 ) = 2

and the random weights Q n are calculated from Q0 =

cos φ yy0 , cos φ y0 y

Q n = Q n−1

cos φ y n−1 y n , cos φ y n y n−1

n = 1, 2, . . . , N t−t = max{n : τ n > 0}. N t−t Denote by ω the random trajectory {(y n , τ n )}n=0 and by Eω the expectation over its distribution. The random variable η yt depends on the parameters x , t and ω. By the construction, we get 



t−t

Eω {η yt |(x , t )} =





∂Z0 (y − y , t − t − τ)μ(y , τ; x )dσ(y ). ∂n(y)

dτ 0

Γ

Consequently, ζ yt =

η yt f (x , t ) p(x , t )

is an unbiased estimator for J3 (y, t): Eζ yt = J3 (y, t).

98 | 4 Walk on Boundary for heat equation

Next, we analyse the variance. By definition,  2    f (x , t ) 2 = E(x ,t ) 2   Eω {η2yt |(x , t )} . Eζ yt p (x , t )

(4.88)

Lemma 4.17. Suppose that the function f is chosen so that f (x, t) ≡ 0,

x ∈ Γδ ,

where Γ δ = {x : x ∈ G; inf |x − y| < δ} y∈Γ

is the δ-neighbourhood of Lyapunov continuous Γ (δ-strip near the boundary). Choose the density p(x , t ) in such a way that t

dt

0



f 2 (x , t )  dx < ∞. p(x , y )

G

Then, the variance of ζ yt is finite. Proof. Indeed, from (4.84), we get sup |μ(y, τ; x )| < c,

x ∈G\Γ δ

where c is a constant that depends on sup |ψ(y, τ; x )|, on the form of G and on δ. From this, we get the uniform estimation: E{η2yt |(x , t )} < C1 ,

(x , t ) ∈ (G \ Γ δ ) × (0, t).

Hence, 2 Eζ yt

t = 0

dt



G\Γ δ

f 2 (x , t ) E {η2yt |(x , t )}dx < ∞. p(x , t )

5 Spatial problems of elasticity 5.1 Elastopotentials and systems of boundary integral equations of the elasticity theory In this chapter we consider differential equations of elasticity theory and use theoretical results given in [58]. Let u(x) be a vector function of displacements, u : G → R3 , u ∈ C2 (G) ∩ C(G), where G is a bounded simple connected domain in R3 . It is well known then that a vector of displacements satisfies the Lamé equations ∆* u(x) ≡ μ∆u(x) + (λ + μ) grad div u(x) = 0,

x ∈ G,

(5.1)

where λ, μ are positive constants that characterize an elastic matter in the domain G. Commonly, these coefficients are called the Lamé parameters. Let Γ = ∂G be a Lyapunov simple connected surface, and φ(y), y ∈ Γ is some continuous vector function on the boundary. Then, we can consider the first BVP for the Lamé equations u(y) = φ(y),

y ∈ Γ,

(5.2)

and the second BVP Tu(y) = φ* (y),

y ∈ Γ,

(5.3)

where T = {T ij }, i, j = 1, 2, 3, is the tension operator of the classical elasticity theory, T ij = λn i (y)

∂ ∂ ∂ + μn j (y) + μδ ij . ∂y j ∂y i ∂n(y)

Here, δ ij is the Kronecker symbol, and n(y) = (n1 (y), n2 (y), n3 (y))T is, as usual, the interior normal vector at a point y ∈ Γ. Vector equation (5.1) is an elliptic system with constant coefficients, and its fundamental solutions can be written out explicitly. One of the classical fundamental solutions is the Kelvin matrix S[x, y] = {s ij (x, y)}, i, j = 1, 2, 3, where s ij (x, y) = λ

δ ij |x − y|

+ μ

(x i − y i )(x j − y j ) , |x − y |3

λ = (λ + 3μ)(4πμ(λ + 2μ))−1 , μ = (λ + μ)(4πμ(λ + 2μ))−1 .

100 | 5 Spatial problems of elasticity

It means that ∆*x S = δ(x − y)E, where E = {δ ij } is the unit matrix. In the full analogy with a scalar case of the Laplace equation, we can introduce vector surface potentials (elastopotentials): a single-layer potential,  V * (x) = S[x, y]Ψ(y)dσ(y), (5.4) Γ

and a double-layer potential,  W * (x) =

K[x, y]Φ(y)dσ(y).

(5.5)

Γ

Here K[x, y] = {k ij (x, y)}, i, j = 1, 2, 3, k ij (x, y) =

δ ij 1 ∂ · 2π ∂n(y) |x − y|   3  δ (x − y )(x − y ) +μ M jk (λ − μ ) ki + 2μ k k 3i i , |x − y| |x − y| k=1

M jk = n k (y)

∂ ∂ − n j (y) , ∂y j ∂y k

k, j = 1, 2, 3.

Performing all differentiations and calculations, we can write out k ij in the explicit form cos φ yx (x i − y i )2  1 1 μ + 3(λ + μ) , · · 2π λ + 2μ |x − y|2 |x − y |2 (x i − y i )(x j − y j ) 3 λ+μ k ij (x, y) = · · cos φ yx · 2π λ + 2μ |x − y |4 n i (y)(x j − y j ) − n j (y)(x i − y i ) μ + , i  = j. · |x − y |3 π(λ + 2μ)

k ii (x, y) =

(5.6)

(s) Denote by k(r) ij the first term in the last sum, and by k ij , we denote the second term. The notation is related to different properties of these kernels when |x − y| → 0, x, y ∈ Γ. On Lyapunov surfaces, we have −2+γ ), k(r) ij (x, y) = 0 (| x − y | −2 k(s) ij (x, y) = 0 (| x − y | ), (s) where γ is the Lyapunov constant. Hence, k(r) ij is a weakly polar kernel and k ij is a singular kernel.

5.1 Elastopotentials

|

101

Theorem 5.1. [58] If densities Φ, Ψ are Hölder continuous vector functions on Γ with exponent equal to β : Φ, Ψ ∈ C0,β (Γ), then (1) elastopotentials V * , W * satisfy the Lamé equation ∆* V * (x) = ∆* W * (x) = 0 at every point x ∈ Γ; (2) the limiting values of W * satisfy the relation (W * (x))± = ∓Φ(x) + W * (x),

x ∈ Γ,

(5.7)

where W * (x) is the direct value of a double-layer potential; (3) the limiting values of the tension operator applied to V * satisfy (TV * (x))± = ±Ψ(x) + TV * (x),

x ∈ Γ.

(5.8)

Following this theorem, we seek a solution to problem (5.1), (5.2) in the form of a double-layer elastopotential W * (x). This leads to the integral equation of the second kind for an unknown vector density, Φ:  Φ(y) = −κ K[y, y ]Φ(y )dσ(y ) + φ(y), κ = 1, (5.9) Γ

or in the operator form Φ = −κK Φ + φ. Consider second BVP (5.3) for Lamé equations (5.1) in G1 and seek its solution in the form of a single-layer elastopotential V * (x). Then, in the full analogy with the classical Neumann problem for the Laplace equation, we come to the integral equation of the second kind for the vector density Ψ of this potential  Ψ(y) = −κ K[y , y]Ψ(y )dσ(y ) + φ* (y), (5.10) Γ

which is adjoint to (5.9). Considering K and K* as integral operators in the space of vector functions 0,β C (Γ), we can assert that they, despite their singularity, have the same spectral properties as the corresponding integral operators (3.6) and (3.22) of the classical potential theory. Hence, the Neumann series for operators K , K* do not converge, but equations (5.9), (5.10) have the unique solutions. These solutions can be found with the help of the same transformations and analytical continuation methods that have been used in Chapter 3 in the case of the BVPs for the Laplace equation. As a consequence, we can write out a solution of the first interior problem for the Lamé

102 | 5 Spatial problems of elasticity

equations in the form of a finite sum u(x) =

n  i=0

l(n) i



K[x, y]K i φ(y)dσ(y) + δ (n) (x),

(5.11)

Γ

where l(n) i are determined by the analytical continuation method (Chapters 2 and 3). It is essential to note that in this case, a proof of the convergence of the transformed series is significantly more complex problem since analyticity of the matrix resolvent for singular integral equation needs its special and careful investigation [58].

5.2 First BVP and estimators for singular integrals Considering first BVP (5.2) for the Lamé equations (5.1), we seek its solution in the form of a double-layer elastopotential W * . As a consequence, we come to the need of computing the iterations for the boundary matrix-integral singular operator, K. It is obvious that every unbiased estimator ξ i or ξ i* (Chapter 2) has infinite moments E|ξ i |m , m ≥ 1 in the case when a kernel of an integral operator is singular. Further in this chapter, we construct an unbiased estimator for a singular part of the integral operator K with finite variance. To do that, we will use the kernel symmetry properties. Consider a singular integral operator acting on functions defined on a Lyapunov surface Γ ∈ R3 ,  (5.12) K (s) φ(y) = k(y, y )φ(y )dσ(y ). Γ

Here, singularity means that |k(y, y )| = O(|y − y |−2 ) for |y − y | → 0. So this integral does not exist in the ordinary sense. By definition, we have  (s) K φ(y) = lim k(y, y )φ(y )dσ(y ), ε→0 Γ (ε)

if the limit in the right-hand side exists. Here, Γ (ε) = Γ \ {y : |y − y | ≤ ε}. It is well known [70] that both necessary and sufficient condition of existence for K (s) φ(y) is the following equality: 2π k0 (y, ψ)dψ = 0, 0

(5.13)

5.2 First boundary value problem

|

103

where k(y, y ) · |y − y |2 , k0 (y, ψ) = lim  y →y

and y approaches y in such a way that ψ = ∠(π(y ) − y, e)

(5.14)

remains constant. Here, we denote by π(y ) the projection of y on the tangent plane at the point y, and e is some fixed direction on this plane. It is obvious that to construct an effective estimator, we have to use condition (5.13). One of the possible ways of its utilizing is the method proposed in the study of Wagner [121]. This algorithm is a variant of a stratified sampling method [23], and it can be applied in the case when we exactly know the region Γ+ (y) where the kernel k(y, y ) is positive, and the corresponding region Γ− (y) where this function is negative. Different techniques will be employed here. It is based on some symmetrical mappings, which are used for constructing walk-on-boundary branching Markov chain. To define these mappings, we introduce the local spherical coordinates with the origin at the point y in such a way that y = (r, θ, ψ) where r = |y − y |, ψ is defined in (5.14) and θ is the angle between the vector y − y and the tangent plane at the point y. So, when y  ∈ Γ, then α y (y ) is defined as such a point on Γ that |y − α y (y  )| = r,

and ∠(π(α y (y ) − y), e) = ψ + π. ¯ ψ + π), where θ¯ = θ(1 + 0(rγ )), as a consequence of the It means that α y (y ) = (r, θ, Lyapunov’s continuity of the boundary. Introduce another symmetrical mapping. Let us define χ y (y ) = (¯r , θ, ψ + π). It means that the angle between the vector χ y (y ) − y and the tangent plane is equal to θ and r¯ is obtained from the condition that χ y (y ) lies on Γ. Obviously, r¯ = r(1 + 0(rγ )) for a Lyapunov boundary. Consider now regular and singular parts of the kernel k ij (y, ·) where the second argument is obtained as a result of a symmetrical mapping applied to y . Without loss of generality, the point y may be considered as either elliptic or an interior point of a plane part of the boundary. In the first case,  (r)  γ k(r) ij (y, α y (y )) = k ij (y, y )(1 + 0(r )),  (s)  γ k(s) ij (y, α y (y )) = −k ij (y, y )(1 + 0(r )),

(5.15)

104 | 5 Spatial problems of elasticity

and hence k ij (y, α y (y )) = −k ij (y, y )(1 + 0(rγ )). The same relations are valid for χ y (y ) also. It is obvious enough that points α y (y ) and χ y (y ) coincide in the case when y, y lie in the same plane and symmetrical mapping does not move us out of it. We have k(r) (y, y ) = k(r) (y, α y (y )) = 0 here, since n(y ) is orthogonal to both y − y and y − α y (y ). Hence,   k ij (y, α y (y )) = k(s) ij (y, α y (y )) = −k ij (y, y ).

(5.16)

Fix now some point y ∈ Γ and construct an estimator for K φ(y) where φ is a Hölder continuous function on Γ. To do that, we have to choose some distribution density for a point y . At first, we consider y to be an elliptic point. In this case, it is more convenient to define the required distribution density not in the surface measure σ but in the angle measure, Ω: dΩ(y ) =

cos φ y y dσ(y ). | y − y  |2

Direction y − y is determined by two angles, θ and ψ. ψ is chosen randomly (uniformly) on the segment (0, 2π) and θ is sampled proportionally to (sin θ)−β for some β ≥ 0. So we have p¯ (β) (θ, ψ) = const · (sin θ)β . Note that if β = 0, then y is isotropically distributed in the solid angle with its vertex at the point y. This means that for β = 0, the distribution density of y in the surface measure, p(β) (y, y ) = p¯ (β) ·

cos φ y y , | y − y  |2

(5.17)

is equal to the transitional density (3.10) of an isotropic WB-process. Hence, we have p(β) (y, y ) = O(|y − y |−2+γ−β ),

(5.18)

when |y − y | → 0. Thus, we have to choose β < γ . Recall that we used such non-isotropic densities in the third and fourth chapters. Suppose now that y is located in the plane part of the boundary. It means that there exists such r0 > 0 that the circle C = {y : y ∈ Γ, |y − y | < r0 }

(5.19)

5.2 First boundary value problem

|

105

is totally contained in the same plane. We define the distribution density in this circle by the formula p(β) (y, y ) = const · |y − y |−1−β ,

(5.20)

for some β ≥ 0. It means that angle ψ is sampled isotropically, and constant factor is chosen in such a way that  p(β) (y, y )dσ(y ) = 1. C

Next, we define a transitional density p(y, y ) by the following procedure. Let ω be a positive number less than 1. With probability ω, we sample y in the neighbourhood C of the point y (see (5.19)) with the density p(β) , defined in (5.17) or (5.20), depending on the geometrical properties of y. With the complementary probability, 1 − ω, point y is chosen randomly in Γ \ C in accordance with some convenient density p1 , which may, for example, be equal to the transitional density of an isotropic WB-process. So we have p(y, y ) = ωp(β) (y, y ) · I c (y ) + (1 − ω)p1 (y, y ) · I Γ\C (y ),

(5.21)

where I C is the indicator function of set C. Recall now that our goal is to construct an estimator for a singular integral, and this is the purpose of introducing the density (5.21). Denote y0 = y, y11 = y and y12 = α y (y ) (or χ y (y )) and consider the estimator   1 K[y0 , y11 ]φ(y11 ) K[y0 , y12 ]φ(y12 ) , (5.22) ζ1* (y0 ) = − + 2ω p (y0 , y12 ) p(β) (y0 , y11 ) if y ∈ C, and ζ1* (y0 ) = −

1 K[y0 , y11 ]φ(y11 ) , · 1−ω p1 (y0 , y11 )

(5.23)

for y ∈ Γ \ C. It means that we use the transitional density (5.21) and bifurcate the trajectory of WB-process {y i } in the case when y i+1 is located in the neighbourhood of the point y i . In (5.22), we denote by p (y0 , y12 ) the distribution density of the point y12 . This point is obtained as the result of some symmetrical mapping applied to y11 and hence, in the general case, its distribution depends on the geometrical properties of Γ in C. If C, for example, is a part of a sphere or of a plane, then p (y0 , y12 ) = p(β) (y0 , y11 ) for both α and χ. By definition, angle ψ (one of the angle coordinates of the point y11 ) is isotropically distributed, and hence ψ + π is also isotropically distributed. As a consequence, angle distributions of the points y11 and χ y (y11 ) coincide. So we have

106 | 5 Spatial problems of elasticity

in this case, p (y0 , y12 ) = p β (y0 , y11 ) ·

cos φ y12 y0 r2 · . cos φ y11 y0 r¯ 2

As for the distribution p , which is considered in the surface measure, σ, we can assert only that p (y0 , y12 ) = p(β) (y0 , y11 ) · (1 + 0(rγ )), where γ is the boundary Lyapunov constant. It is obvious that Eζ1* (y0 ) = K φ(y0 ). Next, we establish that the variance of this estimator is finite. To do that, it is sufficient to consider a singular part of a non-diagonal element of matrix K[y0 , y11 ] and show that an estimator   (s) (s) 1 k ij (y0 , y11 )φ j (y11 ) k ij (y0 , y12 )φ j (y12 ) ξ (y0 ) = + 2 p (y0 , y12 ) p(β) (y0 , y11 ) for the integral,

 Γ

k(s) ij (y 0 , y 1 )φ j (y 1 )dσ(y 1 ), has finite second moment. Suppose that

φ j ∈ C0,s (Γ), i.e., Hölder continuous with the exponent s. Consider the function ζ = ξ 2 (y0 )p(β) (y0 , y11 ) when r = |y0 − y11 | tends to zero. Applying (5.15) and then (5.18), we come to the following relation in the case when y is an elliptic point: ζ=

(s) 2 1 (k ij (y0 , y11 )) · φ2j (y11 ) · 0(r2 min(γ,s) ) = 0(r−2−γ+β+2 min(γ,s) ), · (β) 4 p (y0 , y11 )

while for the case when y is located in a plane part of Γ, we have ζ=

(s) 2 1 (k ij (y0 , y11 )) · φ2j (y11 ) · 0(r2s ) = 0(r−3+β+2s ). · (β) 4 p (y0 , y11 )

Thus, to guarantee the variance of a constructed estimator is finite, the choice of β should be adjusted to the smoothness properties of both Γ and the function under the integral sign. In the elliptic case, we must have γ > β > γ − 2 min(s, γ ), and in the plane case, the following condition should be satisfied: 1 > β > 1 − 2s. As a consequence, the isotropic transitional density can be used for an elliptic point y if s ≥ γ /2 + ε for some ε > 0. In the case when Γ is strictly convex, the isotropic density on the whole boundary can be used. So we see that if we construct the estimator in accordance with formulas (5.22), (5.23) and use the transitional density (5.21) for simulation of WB-process, then on every step, we have * (y0 ), E(ζ i* (y0 )/y0 ) = K ζ i−1

5.3 Other boundary value problems

| 107

and moreover, the conditional second moment of ζ i* is finite. It means that the estimator ζ i* is constructed recurrently according to the following formulas: ζ i* (y0 ) = −

  * * (y11 ) K[y0 , y12 ]ζ i−1 (y12 ) 1 K[y0 , y11 ]ζ i−1 , + 2ω p (y0 , y12 ) p(β) (y0 , y11 )

for y = y11 in the neighbourhood C of the point y0 , and ζ i* (y0 ) = −

* (y ) K[y0 , y11 ]ζ i−1 1 11 , · 1−ω p1 (y0 , y11 )

for y ∈ Γ \ C. The WB-process starts from the point on Γ, y0 , which can be sampled using the initial distribution density p0 from (5.9), that is, isotropically in the solid angle with its vertex at the point x0 ∈ G. We can take x0 = x, the point, in which we want to calculate the solution of BVP (5.1), (5.2). Next, we use the transitional density (5.21) and construct a Markov chain, branching on every step with some probability, ω. Then, we have that ξ i* =

K[x, y0 ] * ζ (y0 ) p0 (y0 ) i

is an unbiased estimator for the iteration of the integral operator, K: Eξ i* = (K, K i φ), and its variance is finite. Substituting ξ i* in (5.11), we obtain a δ(n) -biased estimator for the solution to the BVP, u(x): ξ* =

n 

* l(n) i ξi ,

i=0

where δ(n) (x) is a vector function whose norm is decreasing as n → ∞ at the speed of a geometric progression.

5.3 Other BVPs for the Lamé equations and regular integral equations Consider the second BVP (5.3) for the Lamé equations in the exterior G1 ⊂ R3 of a bounded domain G with a simple connected boundary. We seek its solution in the form of a single-layer elastopotential V * . As a consequence, we come to the integral equation (5.10) for an unknown potential density Ψ. This integral equation is adjoint to (5.9). So in the full analogy with a scalar case, we can use the same WB-process

108 | 5 Spatial problems of elasticity

that has been simulated to obtain an estimator for the first BVP and construct a direct vector estimator. From here we have Eξ i = (K*i φ* , S), and ξ=

n 

l(n) i ξi

i=0

is a δ(n) -biased estimator for u(x) where ξ i = ζ i [x, y0 ]

φ* (y0 ) , p0 (y0 )

ζ0 [x, y0 ] ≡ S[x, y0 ] and the matrix-valued random weights ζ i are obtained from the recurrent relation   1 ζ i−1 [x, y11 ]K[y0 , y11 ] ζ i−1 [x, y12 ]K[y0 , y12 ] , + ζ i [x, y0 ] = − 2ω p (y0 , y12 ) p(β) (y0 , y11 ) if the Markov chain is branching at the point y0 , and ζ i [x, y0 ] = −

1 ζ [x, y11 ]K[y0 , y11 ] , · i−1 1−ω p1 (y0 , y11 )

otherwise. Next, in analogy with the Laplace equation, consider the first exterior and the second interior BVPs for the Lamé equations. If we try to seek their solutions in the form of a double-layer elastopotential W * and a single-layer elastopotential V * , respectively, we come to integral equations (5.9) and (5.10), where κ is taken to be equal to −1. Since κ = −1 is a characteristic value of the operator K, and Fredholm theorems are valid for this operator, then it means that solution of integral equations (5.9), (5.10) for κ = −1 exist if and only if their free terms are orthogonal to all corresponding eigenfunctions of K* and K, respectively. It is known [58] that there are six of them, for both of integral operators. So these orthogonality conditions cannot be effectively utilized in the process of an algorithm construction. The conclusion is that if we want to use a WB-process for solving the first exterior and the second interior BVP, then we have to consider other representations of a solution in the form of surface potentials. The first exterior problem, for example, can be solved with a help of a WB-process, if we seek its solution (by analogy with the Laplace equation) as a weighted sum of W * and V * . It is essential to note that in addition to the classical fundamental solution S, there exist other fundamental solutions of the Lamé equations, and as a consequence,

5.3 Other boundary value problems

| 109

different forms of surface potentials can be used. For example, the surface potential  W1* (x) = K1 [x, y]Φ1 (y)dσ(y) Γ

satisfies the Lamé equation at every point x ∈ Γ and has the same limiting properties when x approaches the boundary. Here, the matrix-valued kernel K1 [x, y] = {k(1) ij (x, y)}, i, j = 1, 2, 3 is defined by the following relations:   cos φ yx 1 xi − yi 1 , · 2μ − 3(λ + μ) · · 2π λ + 3μ |x − y|2 |x − y |2 (x i − y i )(x j − y j ) 3 λ+μ k(1) , i  = j. ij = 2π · λ + 3μ · cos φ yx · |x − y |4 k(1) ii =

(5.24)

Consider the first interior BVP for the Lamé equations and seek its solution in the form of this potential, W1* . As the result, we come to the boundary integral equation of the second kind for an unknown vector function Φ1 :  Φ1 (y) = −κ K1 [y, y ]Φ1 (y )dσ(y ) + φ(y), Γ

κ = 1, y ∈ Γ. It is known that this integral equation has the same spectral properties as (5.9), but the kernel K1 (as we see from (5.21)) does not have singular components. So we can use the same isotropic WB-process that has been constructed in Chapter 3 for solving the Laplace equation. Let {y i , i = 0, 1, 2, . . .} be a trajectory of this Markov chain, p0 its initial and p its transitional densities. Then, we can use a standard adjoint vector estimator for the solution u(x): ξ * (x; n) =

n 

* l(n) i Q i φ(y i ),

i=0

where the matrix-valued weights are calculated using the following formulas: Q*0 =

K1 [x, y0 ] , p0 (y0 )

Q*i = −

Q*i−1 K1 [y i−1 , y i ] , p(y i−1 , y i )

and the coefficients l(n) i are defined by a method of analytical continuation. In conclusion to this chapter, we note that there exist many other elasticity problems, which can be treated in an analogous way. This means that some sort of WB-process and, hence, Monte Carlo estimators can be constructed for their solutions.

6 Variants of the random walk on boundary for solving stationary potential problems 6.1 The Robin problem and the ergodic theorem In all random walk algorithms described in the book up to this point, the solution was represented in the form of an average over a set of independent random trajectories. It is well known, however, that there exists another approach to constructing Monte Carlo algorithms. The methods that exploit such approach date back to the work of Metropolis [67] and are based on a representation of a solution as an average over one long trajectory. Currently, Markov chain Monte Carlo computational methods that stem from this original idea are widely used in various applications (see, e.g. [5, 92]). Further in this section, we describe how this approach, first suggested in [97], can be used for solving the Robin problem. First, consider the classical problem of electrostatics. Usually, it is formulated the following way: to find the electrostatic potential satisfying the Laplace equation ∆φ = 0 outside a system of isolated conductors G i , such that the natural zero conditions at the infinity are satisfied, φ(r) → 0 as r → ∞, and the potential values, φ |Γ i = φ i , are given on the boundaries of these conductors Γ i = ∂G i . In applications, the following condition should be taken into account:  ∂φ dσ = −4πl i , ∂n Γi

where l i is the total charge of the ith conductor. If only one conductor G1 is given, then the solution can be represented as φ = φ1 V(x, y, z), where V is the solution to the first exterior BVP such that V |∂G1 =Γ1 = 1, and φ1 is determined from  ∂φ dσ = −4πl1 , ∂n Γ1

i.e., φ1 = l1 /C, where C=−

1 4π

 Γ1

is the capacitance of the conductor.

∂V dσ ∂n

6.1 The Robin problem and the ergodic theorem

For a system of conductors,  C ij (φ j − φ i ), l i = C ii φ i + j = i

| 111

C ij = C ji , i = 1, . . . , n,

where l i is the charge, and φ i is the potential of the ith conductor. Thus,  1 ∂φ(j) C ij = l(j) = − dσ, i, j = 1, . . . , n, i 4π ∂n Γi

where φ(j) are harmonic functions that satisfy the following boundary conditions: φ(j) |Γ i = 0 (i  = j),

φ(i) |Γ i = 1,

φ(j) |∞ = 0.

The problems with multiply connected geometry will be considered in Section 6.3. Here, we restrict ourselves to a simply connected domain G ≡ G1 and construct an estimate based on ergodic properties of a simulated walk-on-boundary Markov chain. We consider G ∈ R3 to be a compact set representing an electrical conductor. For smooth boundary, Γ ≡ ∂G, there exists the unique solution of the exterior Dirichlet problem we have formulated. It is sufficient, for example, to demand Γ to be a regular piecewise Lyapunov surface [34]. In this case, it is possible to represent V as a single-layer potential  1 μ(y )dσ(y ), V(x) = |x − y | Γ

with a charge density, μ, which is the unknown. V is commonly called the Robin potential. Mathematically, the problem is to compute the integral  C = μ(y) dσ(y) Γ

and the function μ(y) = −

1 ∂V (y). 4π ∂n

(6.1)

To find an equation satisfied by μ, we make use of the well-known jump properties of a single-layer potential’s normal derivative. Hence, we have (for x ∈ R3 \G and y ∈ Γ, y−x | cos φ yx | = | |x−y| · n(y)| ≥ α > 0) ∂V (y) = lim(∇x V(x) · n(y)) ∂n x→y  cos φ yy μ(y )dσ(y ) − 2πμ(y). =− | y − y  |2 Γ

(6.2)

112 | 6 Variants of the Random Walk on Boundary

From (6.1) and (6.2), it follows that  μ(y) =

cos φ yy μ(y )dσ(y ), 2π|y − y |2

Γ

or in the operator notation μ = Kμ.

(6.3)

These equations imply that μ is the eigenfunction of the integral operator, K, corresponding to the maximal eigenvalue, which equals 1 [32]. Suppose that G is convex, then the kernel of the integral operator k(y, y ) = cos φ yy is non-negative and 2π|y−y |2 

cos φ yy dσ(y) = 1. 2π|y − y |2

Γ

This normalization means that the kernel of the adjoint operator, K * , k(y n+1 , y n ) = r(y n → y n+1 ) can be considered as a transition probability density of a homogeneous Markov chain {y n }∞ n=1 of points on Γ. This density function corresponds to the uniform distribution of successive points, y n+1 , in the solid angle with vertex y n . Therefore, the Markov chain we constructed is the 3D version of the isotropic random walk-on-boundary process defined in Section 3.2.  Hence, we can think of Ω(y , ς) = ς k(y, y )dσ(y) as a probability measure defined for every open set ς in Γ. This measure is based on choosing points uniformly in a solid angle subtended by ς and having y as its vertex. If Γ is strictly convex, then the angle measure, Ω, and surface measure, σ, are absolutely continuous, meaning that Ω(y , ς) is strictly positive [26]. It is well known [47] that the weakly singular integral operator, K, is completely continuous. Thus, Ω(y , ς) is regular and we can apply the ergodic theorem to statistics created using this Markov chain [26]. Suppose now that there are planar segments of the boundary. In this case, not every set of non-zero surface area σ(ς) has non-zero angle measure Ω(y , ς) for all points y ∈ ∂G. However, it can be easily shown that the second iteration, K 2 , of the integral operator defines a strictly positive measure Ω(2) (y , ς) =  Ω(y, ς)k(y, y )dσ(y). Thus, we can apply the ergodic theorem in this case as well. ∂G Therefore, by this theorem, there exists a positive stationary distribution, Π∞ , of the Markov chain as defined above. This means that  Π∞ (ς) = Π∞ (dσ(y )) Ω(2) (y , ς) Γ

6.1 The Robin problem and the ergodic theorem

|

113

for every open set ς ⊂ Γ. This also implies that the distribution is absolutely continuous and its density, π∞ , satisfies the equation π∞ (y) = K 2 π∞ (y). Also, the existence of π∞ (x) can be derived using the Döblin condition [55]. This condition is satisfied since Γ is an ergodic class. Hence, since μ = Kμ = K 2 μ, (6.4)

μ = Cπ∞

for some constant C. This constant must equal the capacitance of G, since π∞ is a probability density. Here, we made use of the fact that 1 is a simple eigenvalue of the integral operator K, and there is only one eigenfunction corresponding to this eigenvalue [34]. By the ergodic theorem [38, 68], for an arbitrary initial distribution, Π0 , and bounded function v,  I[v] ≡

1 v(y n ). N→∞ N N

v(y)π∞ (y)dσ(y) = lim

(6.5)

n=1

Γ

Note that we use both even and odd indexed points of the Markov chain in this sum since by (6.4), π∞ = Kπ∞ .

6.1.1 Monte Carlo estimator for computing capacitance Consider the Robin potential, V, inside G. The boundary conditions state that the Robin potential is constant and equal to 1 in G. Thus, we have  1 μ(y )dσ(y ) = 1, (6.6) |x − y | Γ 1 . for any point x ∈ G. Therefore, we may fix x ∈ G and set v(y) = |x−y| Together, relations (6.4), (6.5), (6.6) result in the following formula [104]:

" C=

#−1

1 v(y n ) lim N→∞ N N

,

(6.7)

n=1

which will be used to calculate the capacitance. To estimate the computational error, we use a Markov chain version of the central limit theorem [38, 68]. It states that N1 Nn=1 v(y n ) tends to a normally distributed

114 | 6 Variants of the Random Walk on Boundary random variable with mean I[v] and variance σ2 N −1 . Here, 



2

σ = lim

π∞

N→∞ Γ

2

) 1 ( √ v(y n ) − I[v] N n=1 N

.

This means that error of our computational algorithm is of the statistical nature. Hence, for a given accuracy ε, the cost of computations is of order σ2 ε−2 . To evaluate σ2 , we use the method of batch means [11] with the number of batches, √ √ k + 1, equal to N + 1 and the batch size, m, equal to N. Thus, we have σ2 =

2 k  1 m 1 Si − S , m N m→∞, k→∞ k lim

i=0

k where S i = m(i+1) i=0 S i . j=mi+1 v(y j ), S = Note that the algorithm based on (6.7) provides a method to obtain the value of the capacitance without explicitly calculating the density, μ.

6.1.2 Computing charge density To calculate the charge distribution, we use relations (6.4) and (6.5) and construct Monte Carlo estimators for iterations either of the integral operator K or its adjoint, K * . These estimators are based on a random walk-on-boundary process that is not necessarily isotropic. Let p(y n → y n+1 ) be the transition probability density of this Markov chain, {y n , n = 0, 1, . . .}. Then, for some integrable functions f ∈ L(Γ) and h ∈ L* (Γ), the direct and adjoint estimators, respectively, are defined as [104] (h, K n f ) = EQ n h(y n ) = EQ*n f (y n ). Here Q0 =

f (y0 ) , p0 (y0 )

Q n+1 = Q n

k(y n+1 , y n ) , p(y n → y n+1 )

Q*0 =

h(y0 ) , p0 (y0 )

Q*n+1 = Q*n

k(y n , y n+1 ) . p(y n → y n+1 )

and

Therefore, since we are integrating an absolutely convergent series, we have C (h, K n π0 ). N→∞ N N

(h, μ) = C(h, π∞ ) = lim

n=1

(6.8)

6.2 Stationary diffusion equation with absorption

|

115

It is clear that to compute the density, μ(y), at some point, y ∈ Γ, we have to set h(y0 ) = δ(y − y0 ). Note, however, that this last equality is valid only for bounded functions, h. To overcome this, we introduce a partition, ς j , j = 1, . . . , m, on the  boundary surface: Γ = m j=1 ς j , and use a piecewise constant approximation for μ. Thus, we reduce the problem to the estimation of a finite number of cell values or, in other words, to a finite number of functionals (6.8) with different weight functions h j (y) = χ[ς j ](y)/σ(ς j ). Here χ[·] is the indicator function. Hence, it is possible to use a direct estimator, and set p0 = f = π0 . From this, it follows that for convex G, all weights, Q n , are equal to 1 and so μ(y) ≈ lim

N→∞

Nj C Nσ(ς j )

for

y ∈ ςj ,

(6.9)

where N j is the number of Markov chain points that hit the cell ς j [104]. We thus arrive at an algorithm that makes it possible to calculate both the capacitance and the charge distribution simultaneously. Initially, we randomly choose a point y0 on Γ with a probability density π0 . One of the possible choices for such a cos φ yx density is to set π0 (y) = 2π|y−x| 2 for some x inside the domain G. This means that y 0 is distributed isotropically within the solid angle with vertex x. Next, we simulate a long Markov chain of points using isotropic random walk on boundary and calculate C−1 using (6.7), and the numbers, N j , with the methods described above. Finally, using (6.9), we obtain an approximation to the charge distribution.

6.2 Stationary diffusion equation with absorption In this section, we deal with the following BVP (in R3 , λ ≥ 0) : ∆u(x) − λu(x) = 0,

x ∈ G,

u|Γ = g.

(6.10)

Its fundamental solution, which turns to zero at infinity, is Z(x − y |λ) = −

√ 1 1 exp{− λ|x − y|}. 4π |x − y|

Hence, we can use the double-layer potential representation (for the external normal vector, n ξ )  √ √ cos(ξ − x, n ξ ) exp{− λ|x − ξ |}(1 + λ|x − ξ |)dσ(ξ ). (6.11) u(x) = μ(ξ ) 2π|x − ξ |2 Γ |λ) Here, the kernel equals −2 ∂Z(x−ξ and μ is determined from ∂n ξ  μ(y) = − r(y → ξ )h(y, ξ )μ(ξ )dσ(ξ ) + g(y). Γ

(6.12)

116 | 6 Variants of the Random Walk on Boundary

In (6.12), the known functions are √

√

h(x, ξ ) = exp{− λ|x − ξ |}(1 + λ|x − ξ |) q(x, ξ ) sign(cos(ξ − x, n ξ ))

(6.13)

and r(x → ξ ) =

| cos(ξ − x, n ξ )| , 2π|x − ξ |2 q(x, ξ )

(6.14)

where q(x, ξ ) is the number of intersections of the boundary by the line ξ − x. Thus, we construct the Markov chain with the transition density (6.14) (the same function can be used to simulate the initial point of the Markov chain). Note that for sufficiently large λ, the kernel in (6.12) is sub-stochastic. Hence, an unbiased estimator can be constructed if, e.g. sup |h(x, ξ )| < 1.

(6.15)

x,ξ ∈Γ

Condition (6.15) is satisfied if exp(−θ)(1 + θ) < where θ =

√

1 , qmax

λd, d is the domain diameter and qmax = sup q(x, ξ ). x,ξ ∈Γ

For other variants of using the estimator for a non-convex boundary, see the remark in Section 3.7.1.

6.3 Multiply connected domains In this section, we treat BVPs for multiply connected domains, in which the potential theory is more complicated than in the simply connected case. Following the book [34], we consider three cases: (A) G is a bounded simply connected (1-connected) domain, (J) G is a bounded domain, which is represented as G = G0 \

k 2

Gl ,

Gi

3

G j = ∅,

i  = j,

l=1

where all the domains {G i }ki=0 are bounded, simply connected, and G i ⊂ G0 , i = 1, . . . , k. Thus, the boundary of the domain G is (k + 1)-connected: Γ=

k 2 i=0

Γi .

6.3 Multiply connected domains

| 117

(H) G is unbounded: G = R3 \

k 2

Gl ,

Gi

3

G j = ∅,

i  = j.

l=1

In this section, we denote by n the normal vector exterior to G. In Chapter 3, we constructed the walk-on-boundary algorithms for solving the Dirichlet, Neumann and third BVPs (interior and exterior), while in Section 6.1 we described the algorithm of solving the Robin problem for the case (A). We shall generalize these algorithms to the case (J) and (H). In the case (J), as in (A), the condition  g(y)dσ(y) = 0 (6.16) Γ

is necessary to have the unique solution of the interior Neumann problem: ∆φ(x) = 0, x ∈ G, ∂φ → g(y), as y → Γ. ∂n

(6.17)

In the case (J), the resolvent of the boundary integral equation for the density of the double-layer potential has, in general, two poles: λ = 1 and λ = −1. If (6.16) is satisfied, the pole λ = −1 disappears. Thus, the radius of convergence of the series representing the single-layer potential  μ(y)dσ(y) 1 , W(x) = − 2π |x − y| Γ

∂W = −g, ∂n

(6.18)

is equal to 1. Let V(x) = −W(x). Then, W = V1 + (−λ)V2 + (−λ)2 V3 + . . . + (−λ)k−1 V k + . . . ,

(6.19)

where 1 Vk = − 2π



μ k−1 (y)dσ(y) , |x − y|

(6.20)

Γ

1 μ k (y) = − 2π

 Γ

μ k−1 (y )

cos φ yy dσ(y ). | y − y  |2

(6.21)

118 | 6 Variants of the Random Walk on Boundary

The multiplication method (we have already used it in Chapter 3) gives 1 W = − {V1 + (V2 + V1 ) + (V3 + V2 ) + . . . + (V n + V n−1 ) + . . .}. 2

(6.22)

This expansion is then calculated by the walk-on-boundary algorithm as described in Chapter 3. In the case (E), the interior Neumann problem has a solution if  g(y)dσ(y) = 0, l = 1, 2, . . . , k. (6.23) Γl

If this condition is satisfied, the poles λ = 1, λ = −1 both disappear. Hence, the radius of convergence of series (6.19) is larger than 1. Taking in this series λ = −1, we get the solution to the interior BVP: V =−

∞ 

Vj .

j=1

Consider the exterior Neumann problem. The case (E) is treated exactly as the case (A); hence, V=

1 {V − (V2 − V1 ) + (V3 − V2 ) + . . . + (−1)n−1 (V n − V n−1 ) + . . . }. 2 1

(6.24)

In the case (J), the function W has, generally, two poles: λ = 1, λ = −1. However, condition (6.23) implies that the pole λ = 1 disappears; hence, in this case, (6.24) is also true. Let us consider now the Robin problem. The case (J) is treated as the case (A) described in Section 6.1. Indeed, μ(y) = 0 on Γ l , l = 1, . . . , k. Thus, the function μ is defined on Γ0 . The most complicated situation with the Robin problem arises in the case (E). Indeed, for λ = −1, we have k different eigenfunctions satisfying the conditions: μ(l) = −K * μ(l) ,

 μ

(l)

l = 1, . . . , k,

dσ = δ il ,

Γi

where δ il is the Kronecker symbol, i, l = 1, . . . , k. Let  ∂V dσ = α(l) ∂n Γl

(6.25)

6.3 Multiply connected domains

| 119

be the ‘charge’ on the surface Γ l , l = 1, . . . , k. The solution to the Robin problem is fully defined by the expression k 

α(i) μ(i) ,

i=1

where μ(i) , i = 1, . . . , k are found from (6.25). Thus, we have to solve the problem of finding k eigenfunctions of the operator K * corresponding to the eigenvalue λ = −1. Here, we need the information on the geometrical multiplicity of the pole λ = 1. The following result is known [42]. Consider the general case when the domain consists of p non-overlapping domains Ω1 , . . . , Ω p with cuts Φ1 , . . . , Φ q (q is the total number of cuts). Proposition 6.1. The geometric multiplicity of the eigenvalue λ = −1 is equal to p. Functions θ j ∈ C(Γ) defined by θ j = 1 on ∂Ω j and θ j = 0 on ∂Ω k for all k  = j represent the basis of the eigenspace of the operator K (i.e., of the space N(I − K)). For q > 0, λ = 1 is an eigenvalue of K with the geometrical multiplicity q. The functions θ j ∈ C(Γ) defined by θ j = 1 on ∂Φ j and θ j = 0 on ∂Φ k for all k  = j represent the basis of the eigenspace N(I − K). Thus, for the case (E), we take p = k, q = 0. To satisfy the condition (6.25), we choose θ j = 1/S j on ∂Ω j and θ j = 0 on ∂Ω l (l  = j), j = 1, . . . , k, where S j = σ(∂Ω j ) is the area of the surface ∂Ω j . Write down the expansion of the resolvent R(λ, K * ) into the Laurent series in the neighbourhood of the pole λ = λ j = −1: R(λ, K * ) =

∞ 

(λ − λ j )k T kj +

k=0

∞ 

(λ − λ j )−k B kj

(6.26)

k=1

and suppose that B sj θ j  = 0,

B s+1,j θ j  = 0,

(6.27)

where θ j , j = 1, . . . , k are particular initial densities. Let 1  −s+1 −i * i i λ j (K ) θ j , n n

μ jn =

λ j = −1, j = 1, . . . , k.

(6.28)

i=1

Proposition 6.2. Sequence (6.28) converges in norm to the eigenvector μ (j) , which corresponds to the eigenvalue λ j , (i.e., K * μ(j) = λ j μ(j) ), if the finite number of poles of the resolvent R(λ, K * ) lies on the boundary of the spectrum of the operator K * .

120 | 6 Variants of the Random Walk on Boundary

This result is proved [63] in the general case, when the poles λ1 , . . . , λ k have the multiplicities q1 , . . . , q k , respectively. In this case, it is supposed in (6.27) that s ≥ q j , j = 1, . . . , k. Thus, to find numerically the eigenfunctions {μ(i) }ki=1 , we can use the sequence 1  −k i (−1)−i (K * )i θ j , n n

μ(j) ≈ μ jn =

j = 1, . . . , k,

(6.29)

i=1

where the iterations (K * )i θ j are calculated by the walk-on-boundary algorithm, as described for the non-convex case. The rate of convergence of the series is determined by the quantity ln(n)/n. To complete the considerations of the Robin problem, it remains to study the case when then pole is λ = 1 (this appears in the case (J) for the operator K * ). In the case (J), there are k eigenfunctions (Statement 6.1): ν(j) = K * ν(j) ,

j = 1, . . . , k.

Since there are only two simple poles λ = 1 and λ = −1, the radius of convergence of the series (1 − λ2 )ν = ν1 + λ2 + λ2 (ν2 − ν0 ) + . . . + λ n (ν n − ν n−2 ) + . . . is larger than 1. Here, the coefficients {ν n }∞ n=0 are defined by the expansion: ν = ν0 + λv1 + λ2 ν2 + . . . , hence, ν0 = g, ν1 = −K * g, . . . , ν n = −K * ν n−1 . This implies that the sequences ν0 , ν2 , . . ., and ν1 , ν3 , . . ., converge to limits, say H e and H0 , respectively. Then [34] H e − H0 is an eigenfunction corresponding to the eigenvalue λ = 1, and   (H e − H0 ) dσ = 2 g dσ. Γl

Γl

This equality shows that it is possible to find k functions of the type H e and H0 depending on the functions g l : g l (x) = 0, x ∈ Γ l , 1 g l (x) = (l) , x ∈ Γ l , l = 1, . . . , k. 2S

(6.30)

6.3 Multiply connected domains

Let ν(l) = H e(l) − H0(l) . Then,



ν(l) dσ = 0,

Γm



ν

(l)

l  = m; 

dσ = 1;

Γl

| 121

g (l) dσ =

1 . 2

(6.31)

Γm

An arbitrary choice of g generates an eigenfunction of the form ν = α(1) ν(1) + . . . + α(k) ν(k) , where α(l = 2

 g dσ,

l = 1, . . . , k.

Γl

Consider now the interior Dirichlet problem. In the general case of a multiply connected domain, the walk-on-boundary algorithm presented in Chapter 3 does not work since the spectral properties of the integral operator are different. However in the case (E) it works, since the function W = W1 + λW2 + λ2 W3 + . . .

(6.32)

may have only the pole λ = −1. In (6.32),  cos φ yy 1 μ n−1 (y ) dσ(y ), n = 1, . . . , Wn = 2π | y − y  |2 Γ  cos φ y y 1 μ0 = g, . . . , μ n = μ n−1 (y ) dσ(y ), n = 1, . . . . 2π | y − y  |2

(6.33)

Γ

Thus, (6.32) gives the representation for the double-layer potential  cos φ yx 1 W(x) = μ(y) dσ(y) 2π |x − y |2 Γ

with the boundary condition W |Γ = g. To calculate (6.32) at λ = 1, we use the multiplication method described in Chapter 3 (multiplying by (λ + 1)). This gives W=

1 [W − (W2 − W1 ) + (W3 − W2 ) − . . .], 2 1

W |Γ = g.

(6.34)

Let us consider now the interior Dirichlet problem for the case (J). The solution of the BVP ∆V = 0,

V |Γ = g

122 | 6 Variants of the Random Walk on Boundary

is constructed in the form V = W + V0 , where W is the double-layer potential and V0 is the single-layer potential. It is clear that the solution cannot be found in the form of only double-layer potential since λ = 1 is also a pole of W. Although the integral equation μ = λKμ + g may, generally, have λ = 1 as a pole, it is possible to impose some conditions that lead to disappearance of this pole. These conditions are  gν(l) dσ = 0, l = 1, . . . , k, (6.35) Γ

where ν(l) , l = 1, . . . , k, are the eigenfunctions of the equation ν = λK * ν corresponding to λ = 1 (we discussed the construction of these eigenfunctions above, see (6.31)). It is clear that these conditions are artificial for the Dirichlet problem. If in some special case these conditions were satisfied, we could use representation (6.34), since only the pole λ = −1 could exist. Thus, we consider the general case that does not require conditions (6.35). As mentioned, we seek the solution as a sum of double- and single-layer potentials. Let us define a function α(x), x ∈ Γ: α = 0 for x ∈ Γ0 , and α = α(l) on Γ l , l = 1, . . . , k; here, α(l) are some constants, which we choose so that the function g − α satisfies (6.35). From this, we get  α(l) = gν(l) dσ. (6.36) Γ

We can find coefficients γ1 , . . . , γk so that on Γ j , j = 1, . . . , k, the following equalities are true: k  l=1

 γl

ν(l) (y) dσ(y) = α(j) . |x − y|

(6.37)

Γ

Let V0 =

  k Γ

l=1

γl ν(l)

dσ(y) |x − y|

.

(6.38)

Let W(x) be a double-layer potential with the density μ satisfying the equation μ(y) = λKμ + g − α(y),

λ = 1.

(6.39)

In this case, W(x) can be found from (6.3), but W |Γ = g − α. Thus, V = W + V0 , V |Γ = g − α + α = g. Thus, we first calculate α (l) in (6.36) on the basis of (6.29). Then, from (6.37), we find the coefficients γ1 , . . . , γk . The linear functionals in (6.38) are calculated from the

6.3 Multiply connected domains

| 123

approximation (6.28), which in our case takes the form 1 k * i (i) (K ) g j . n n

ν(j) ≈ ν jn =

i=1

We turn now to the exterior Dirichlet problem. The case (E) (including k = 1, i.e., the case (A)). The situation is the same as in the case (J) of the interior Dirichlet problem. We first write down the conditions, which ensures that λ = −1 is not a pole of the density μ determined from μ = λKμ − g.

(6.40)

Since we have already treated the Robin problem for the case (E) and have found k eigenfunctions from (6.25) at λ = −1, we can write the condition which ensures that (6.40) has a solution for λ = −1:  gμ(l) dσ = 0, l = 1, . . . , k. (6.41) Γ

In the case (A), k = 1 and there is one eigenfunction μ(1) . Consequently, if (6.41) is satisfied, the functions μ(l) have no pole at λ = −1. But λ = 1 is also not a pole of μ(l) ; hence, the Neumann series W = W1 + W2 + . . . is convergent. However, conditions (6.41) are artificial, and we use the same approach, constructing the solution in the form of a sum of double- and single-layer potentials. Using the solvability conditions, we choose the function α (as α(l) = const on Γ l ) so that  (6.42) α(l) = gμ(l) dσ l = 1, . . . , k Γ

and the coefficients γ1 , . . . , γk are found from k 

 γl

l=1

μ(l) (y) dσ(y) = α(j) |x − y|

(6.43)

Γ

on Γ j , j = 1, . . . , k. Let V0 =

  k Γ

l=1

γl μ (l)

dσ(y) |x − y|

.

(6.44)

Then, V = V0 − W is the solution of our problem, since V |Γ = α − α + g = g. Linear functionals of the eigenfunctions can be calculated by the scheme (6.29).

124 | 6 Variants of the Random Walk on Boundary

The exterior Dirichlet problem for the case (J). This problem is equivalent to the problem of finding the solution to the exterior Dirichlet problem for the domain with the surface Γ0 and k interior Dirichlet problems in domains with the surfaces Γ l , l = 1, . . . , k. However, we can treat the problem using the general considerations. In this case, we have two poles: λ = −1 and λ = 1 whose geometrical multiplicity is 1 and k, respectively. Hence, the equation adjoint to (6.40) has one eigenfunction μ(1) (we studied this function when constructing the solution to the Robin problem for the case (A)). Consequently, (6.40) has a solution for λ = −1, if  gμ(1) dσ = 0. (6.45) Γ

Thus, if this condition is satisfied, W has only one pole λ = 1; therefore, the multiplication method gives W=

1 [W + (W2 + W1 ) + (W3 + W2 ) + . . .]. 2 1

(6.46)

Conditions (6.45) are artificial for the problem under study. Suppose that  μ(1) dσ = 1. Γ

The potential  C=

μ(1) dσ |x − y|

Γ

is constant inside the domain G. Let



α=

gμ(1) dσ.

Γ

We construct w, the solution of the Dirichlet problem with the boundary conditions: g  = g − α. This function satisfies the condition (6.45); therefore, w=

1 [w + (w2 + w1 ) + (w3 + w2 ) + . . .], 2 1

where 1 w1 = 2π

 (g − α)

cos φ yy dσ(y ), | y − y  |2

Γ

and wk =

1 2π

 w k−1 Γ

w|Γ = g,

cos φ yy dσ(y ). | y − y  |2

6.3 Multiply connected domains

| 125

Therefore, w k = W k in G l and W |Γ = α − g. Let  (1) α μ dσ , V0 = C |x − y| Γ

and V = V0 − W .

(6.47)

This function solves the problem, since V |Γ =

αC − α + g = g. C

The linear functionals of μ(1) such as α, C and 

μ(1) dσ |x − y|

Γ

can be calculated by the algorithm described in Section 6.1. In conclusion, we summarize all the situations we considered in this section. (1) The exterior Neumann problem  ∂φ  ∆φ = 0, = g* . ∂n Γ The boundary integral equation for the density of the single-layer potential is μ* = λK * μ* + g * ;

(λ = 1).

(6.48)

(2) The interior Neumann problem ∆φ = 0,

 ∂φ  = g* . ∂n Γ

The boundary integral equation for the density of the single-layer potential is μ* = λK * μ* − g * ;

(λ = −1).

(6.49)

(3) The interior Dirichlet problem ∆φ = 0,

φ|Γ = g.

The boundary integral equation for the density of the double-layer potential is μ = λKμ + g;

(λ = 1).

(6.50)

126 | 6 Variants of the Random Walk on Boundary

(4) The exterior Dirichlet problem ∆φ = 0,

φ|Γ = g.

The boundary integral equation for the density of the double-layer potential is μ = λKμ − g;

(λ = −1).

(6.51)

(5) The eigenvalue problem for the operator K * at λ = −1: μ(l) = −K * μ(l) .

(6.52)

The geometrical multiplicity of the pole λ = −1 is equal to 1 in cases (A) and (J); the corresponding eigenfunction μ(1) solves the Robin problem. In the case (E), the geometrical multiplicity of the pole λ = −1 is equal to k (k is the number of components of the domain). The corresponding eigenfunctions {μ(l) }kl=1 can be found from (6.29). When solving the exterior Dirichlet problem in cases (A) and (J), only the function μ(1) used, while in the case (E) all functions {μ(l) }kl=1 are involved, since the equation (6.49) is adjoint to (6.51); This leads to the representation of the solution in the form (6.47) (cases (A) and (J)) or (6.42),(6.44) (case (E)). (6) The eigenvalue problem for the operator K * at λ = 1: μ(l) = K * μ(l) .

(6.53)

There is a pole λ = 1 only in the case (J). The geometrical multiplicity of the pole is equal to k, the number of cuts in the domain. Since the equation (6.48) is adjoint to (6.50), then, when solving the interior Dirichlet problem in the case (J), the functions {ν(l) }kl=1 are used.

6.4 Stabilization method To extend the walk-on-boundary algorithms to multiconnected domains, there exists a different approach. The algorithms constructed in Chapter 4 for the heat equation are valid in this general case. We can use these algorithms and then apply the stabilization method. Consider the following pair of equations in a bounded domain: ∂u (x, t) = ∆u(x, t) + f (x), ∂t u(x, 0) = 0, u(y, t) = ψ(y),

x ∈ G, t > 0,

x ∈ G, y ∈ ∂G, t > 0

6.4 Stabilization method

| 127

and ∆v(x) + f (x) = 0, v(y) = ψ(y),

x ∈ G,

y ∈ ∂G.

¯ The function u(x, t) = u(x, t) − v(x) solves the problem: ∂ u¯ ¯ (x, t) = ∆ u(x, t), ∂t ¯ 0) = −v(x), u(x, ¯ t) = 0, u(y,

x ∈ G, t > 0, x ∈ G,

y ∈ ∂G, t > 0.

∞ Denote by {λ k }∞ k=1 the set of eigenvalues and by { φ k }k=1 the eigenfunctions of the Laplace operator:

∆φ k + λ k φ k = 0, φ k (y) = 0,

x ∈ G,

y ∈ ∂G, k = 1, 2, . . .

and λ1 is the minimal eigenvalue. From the representation ¯ u(x, t) = ¯ t) = ∆ u(x,

∞  k=1 ∞ 

c k exp(−λ k t) φ k (x), d k exp(−λ k t) φ k (x),

k=1

where c k , d k are the Fourier coefficients in the expansions of v(x) and f (x), respectively, we get the inequalities ¯ t)||L2 (G) ≤ ||v||L2 (G) exp(−λ1 t), ||u(·, ¯ t)||L2 (G) ≤ ||f ||L2 (G) exp(−λ1 t). ||∆ u(·, ¯ Using the condition u(y) = 0 for y ∈ ∂G and the embedding theorem, we get the following estimation in the uniform norm: ¯ t)||C(G) ≤ c exp(−λ1 t). ||u(·, Consequently, if we solve the stationary problem by the stabilization method and the desired accuracy is defined by the inequality: ¯ t)||C(G) ≤ ε, ||u(·,

128 | 6 Variants of the Random Walk on Boundary (i) then it is necessary to achieve the times t ∼ | ln(ε)|. Since EN t ∼ t, and Dξ xt ∼ t 2 , we conclude that the cost of the stabilization method is

T ε ∼ | ln(ε)|3 /ε2 .

6.5 Nonlinear Poisson equation In contrast to the WOS method, the walk-on-boundary algorithms give the solution at an arbitrary set of points, say, in the points of a grid. This can be used to construct an effective implementation of the known iterative method [12]. Consider the following BVP in a bounded domain G ⊂ Rn : ∆ u = f (x1 , x2 , . . . , x n , u) = f (x, u), u|Γ = φ,

(6.54)

where it is assumed that (1) f has continuous derivatives for all x ∈ G ∪ Γ and for all u; (2) |f (x, u)| ≤ N. Under these conditions, (6.54) has a solution that can be found by the following iterative process. In what follows, we suppose that φ = 0; this can be achieved by adding a harmonic function. Let   ∂f (x, u) k = sup ∂u for all x ∈ G. Denote by v the solution to the BVP: ∆v = −N, v|Γ = 0.

(6.55)

Consider the iterative process: u0 = v, ∆u m − ku m = f (x, u m−1 ) − ku m−1 ≡ g(x, u m−1 ), u m |Γ = 0.

(6.56)

It is known [12] that u m → u monotonically. If G is small, the solution to (6.54) is unique. Generally, (6.56) gives the maximal solution to (6.54). Note that if we start with u0 = −v, we arrive at the minimal solution.

6.5 Nonlinear Poisson equation

|

129

Thus, it is necessary to successively solve the series of BVPs of the type: ∆u m − ku m = g(x, u m−1 ), u m |Γ = 0.

(6.57)

Let Ek (x, y) be the fundamental solution to the equation ∆u − ku = δ(x − y), i.e., √

1 exp[− k|x − y|] Ek (x, y) = . 2π |x − y| We seek the solution to (6.57) in the form  u(x) = g(x, u)u δ (x, y)dy, G

where u δ (x, y) = Ek (x, y) + w(x, y). Hence, w(x, y) solves the problem (∆ x − k)w(x, y) = 0, w(x, y)|x∈Γ = −Ek (x, y)|x∈Γ . First, we write down the random estimator for the problem ∆u m − ku m = 0, u|Γ = φ.

(6.58)

Let ξ (x) be a random estimator for the Dirichlet problem ∆v = 0, v|Γ = φ.

(6.59)

Then, the random estimator for (6.58) is η(x) = Qξ (x), where √

√

Q = (1 − k|y i+1 − y i |) exp[ k|y i+1 − y i |],

130 | 6 Variants of the Random Walk on Boundary

and y i+1 , y i are two successive boundary points of the walk-on-boundary trajectory. Next, to evaluate the solution using its integral representation, we apply the double randomization principle. Thus, the algorithm for solving the nonlinear problem (6.54) can be implemented according to the following scheme: (1) First, store a set of points of the walk-on-boundary process for M trajectories. M should be sufficiently large to attain the desired accuracy. (2) Calculate u0 = v, the solution to (6.55), for all points of a grid chosen in G using the global walk-on-boundary algorithm described in Chapter 3. This is possible since v|Γ = 0. (3) Using the estimator η, perform the same calculation for (6.57). Note that in all estimators, one and the same set of points sampled in p.1 is used. If f (x, u) is a bounded function, we get a Monte Carlo estimator with a finite variance.

7 Splitting and survival probabilities in random walk methods and applications In this chapter we present random walk–based methods for solving some practically interesting problems that are based on the probabilistic representations and different Wiener process properties. In particular, the first passage time and survival probabilities of the Wiener process are used in the problem of diffusion imaging and cathodoluminescence studies. We describe a series of extremely fast stochastic algorithms based on exact representations for the first passage time and exit point probability densities, splitting and survival probabilities. We discuss possible applications of the developed algorithms to the following problems: (1) simulation of epitaxial nanowire growth and (2) diffusion imaging of microstructures, in particular, cathodoluminescence imaging for threading dislocations. In the next chapter, we will use these ideas to simulate the annihilation of electrons and holes in vicinity of nonradiative centres and the quantum efficiency evaluation. In the last example, the random WOS method is used to solve nonlinear diffusion equations and more general systems of nonlinear Smoluchowski equations combined with the kinetic Monte Carlo (KMC) method.

7.1 Introduction The probabilistic interpretations of PDEs and related stochastic algorithms are well developed mainly for parabolic and elliptic partial differential equations (e.g. see [19, 29, 98]). Due to deep intrinsic relation between diffusion random processes and the parabolic equations, the solutions to the relevant BVPs are represented in the form of expectations of functionals over trajectories of the random diffusion processes. A straightforward application of this approach is based on a numerical solution of stochastic differential equations governing the random diffusion processes (e.g. see [44, 52]). Of particular interest are special cases where the solution to a PDE can be represented not through the whole continuous trajectory of the diffusion process but through much simpler random events only, like a survival probability in a domain, the first passage time, known as the exit time from a domain, splitting probabilities describing the probabilities to hit different parts of the boundary, the exit positions and many others. A well known and widely used is the random WOS method for solving linear PDEs with constant coefficients (e.g. see [24, 98]). This technique is extremely efficient when not all the whole solution field is desired but rather some integrals of the solution, or solutions at some prescribed points. In this and the next chapter, we suggest different random walk–based algorithms for solving three classes of physical problems, in particular, (1) a diffusion tomography related to cathodoluminescence imaging technique, (2) simulation of epitaxial growth

132 | 7 Splitting and survival probabilities

of nanowires, and (3) a photoluminescence problem, which is governed by a system of nonlinear diffusion equations for electrons and holes in a semiconductor in the vicinity of defects like dislocations. In these algorithms, we use mainly the survival probabilities, the first passage time and splitting probabilities of the Wiener process. Note that some technical sections present numerous analytical formulas for the splitting probabilities and first passage time and exit position densities for some relatively simple geometrical objects. Although some of these formulas look rather cumbersome like infinite series involving Bessel functions, they are well suited for both analytical and numerical computations.

7.2 Survival probability for a sphere and an interval Let us consider a diffusion problem with a volume absorption of a rate λ2 in a ball B(0, R) centred at the origin: D∆u(x) − λ2 u(x) = 0

x ∈ B(0, R)

(7.1)

satisfying the Dirichlet boundary conditions: u(x) = g(x), x ∈ S(0, R) where S(0, R) is the sphere, the boundary of B(0, R). Let us consider a standard Wiener process W(t) (D = 1) starting from the centre of the sphere. Probabilistic representation of the solution, in any domain, and in particular in B(0, R), has the form u(0) = exp{−λ2 τ} g(W(τ),

(7.2)

where τ is the first passage time of W in B(0, R), and W(τ) ∈ S(0, R) is the exit point of the Wiener process, which is uniformly distributed on the surface of S(0, R). Here, the angle brackets stand for the mathematical expectation with respect to W(t). On the other hand, the Green formula yields [98]  λR u(0) = g(y)dS(y). (7.3) sinh(λR) S(0,R)

Let us take g ≡ 1, then, by comparison with representation (7.2), we obtain exp{−λ2 τ} =

λR , sinh(λR)

(7.4)

which is, due to (7.3), the explicit representation for the survival probability of the Wiener process starting at the centre of a sphere. Let us obtain this result by a direct probabilistic arguments, without appealing to Green formula (7.3). Moreover, we obtain a more general result, namely, we obtain the survival probability of a Wiener process starting from an arbitrary point inside the ball. The proof is based on the use of probabilistic representation (7.2), which is

7.2 Survival probability for sphere and interval

| 133

true for any point inside the sphere. Thus, we are going to evaluate the expectation exp{−λτ} over the Brownian particles starting from an arbitrary point x0 inside the

ball B(0, R) with the first passage time τ. In what follows, we assume, without loss of generality, that D = 1/2, because this can be always adjusted by the relevant normalization of the equation. This is equivalent to change of the absorption rate λ2 to λ2 /2. The probability distribution function of τ has the following explicit form in dimension m (see [116] and [90]): H(t, r, R) =

∞  2R ν J ν (μ n r/R) n=1

r ν μ n J ν+1 (μ n )

exp {−μ2n t/2R2 },

(7.5)

where μ n , n = 1, 2, . . . are positive roots of the Bessel function J ν (z), with ν = m/2 − 1; r is the distance from the starting point x0 to the centre of the /sphere. In our case of dimension 3, we have ν = 1/2, J ν (z) = /  2 sin(z) n = 1, 2 . . .. Besides, J3/2 (z) = πz z − cos(z) . Therefore,

2 πz

sin(z) and μ n = πn,

2R ν J ν (μ n r/R) 2R =− sin(πrn/R)(−1)n . πrn r ν μ n J ν+1 (μ n ) This simplifies the form of the distribution function in dimension 3: H(t, r, R) = −2

∞ 

(−1)n

n=1

R sin(πrn/R) exp {−π2 n2 t/2R2 }. πrn

(7.6)

So by differentiating with respect to t the distribution 1 − H(t, r, R), we get the probability density in the form p(t, r, R) =

∞ 

(−1)n+1

n=1

πn sin(πrn/R) exp {−π2 n2 t/2R2 }. rR

(7.7)

Note that for the Wiener process starting in the centre of the sphere, we get p(t, 0, R) =

∞ 

(−1)n+1

n=1

π2 n2 exp {−π2 n2 t/2R2 }. R2

(7.8)

Now we are ready to evaluate the expectation exp{−λ2 t}. Recall that the density p is true for D = 1/2, which implies, as mentioned above, we should change the absorption rate λ2 to λ2 /2. Let us follow the calculations ∞

2

exp{−λ τ/2} =

dt exp{−λ2 t/2}p(t, r, R)

0

∞ = 0

dt exp{−λ2 t/2}

∞  πn (−1)n+1 sin(πrn/R) exp {−π2 n2 t/2R2 } rR n=1

134 | 7 Splitting and survival probabilities    2  ∞  λ πn π2 n2 = (−1)n+1 sin(πrn/R) dt exp −t + rR 2 2R2 ∞

n=1

0

  ∞  πn 2 (−1)n+1 = sin(πrn/R) 2 rR λ + π 2 n2 /R2 n=1

=

sinh(λr)/λr . sinh(λR)/λR

(7.9)

The last equality follows from the series representation (see [85], 8, p. 731): ∞  (−1)n+1 n=1

n π sin(nx) = sinh(ax)/ sinh(πa). 2 n2 + a2

One-dimensional case. In what follows, we will need also the survival probability for the one-dimensional (1D) case. Let us consider an interval [−R, R], and we are interested in finding the survival probability of a Wiener process starting from an arbitrary point x ∈ [−R, R]. From the general expression for the first passage time, in 1D case, we have π  (−1)k (2k + 1) cos [(2k + 1)/2R] exp{−(2k + 1)2 π 2 t/8R2 }. 2R2 ∞

p(t, x, R) =

k=0

Therefore, ∞

2

exp{−λ τ/2} =

dt exp{−λ2 t/2}p(t, r, R)

0

∞

λ2 t π  } (−1)k (2k + 1) cos [(2k + 1)/2R] 2 2R2 ∞

dt exp{−

=

k=0

0

  (2k + 1)2 π 2 t × exp − 8R2 ∞ cos [(2k + 1)/2R] 4 (−1)k (2k + 1) = π (2k + 1)2 + 4R2 λ2 /π2 k=0

cosh(λx) = cosh(λR) since (see [85], 27, p. 734) ∞  k=0

(−1)k

π cosh(xz) 2k + 1 cos [(2k + 1)x] = . 4 cosh(πz/2) (2k + 1)2 + z2

7.3 The reciprocity theorem

| 135

Two-dimensional case. From the general expression for the first passage time, in 2D case, we have p(t, r, R) =

∞  μ n J0 (μ n r/R) exp{−μ2n t/2R2 }, R2 J1 (μ n ) n=1

where μ n are the positive solutions of the equation J0 (x) = 0. Therefore, exp{−λ2 τ/2} =

∞

dt exp{−λ2 t/2}p(t, r, R)

0

∞

n=1

0

=

∞  n=1

=

μ n J0 (μ n r/R) R2 J1 (μ n )

∞  J0 (μ n r/R) n=1

=

λ2 t  μ n J0 (μ n r/R) } exp{−μ2n t/2R2 } 2 R2 J1 (μ n ) ∞

dt exp{−

=

J1 (μ n )

∞

exp{−[t(μ2n /2R2 + λ2 /2)]} dt

0

2μ n R2 λ2 + μ2n

!

I0 (λr) , I0 (λR)

(7.10)

where I0 (x) is the zero-order modified Bessel function: I0 (x) = J0 (ix). In the last equality in (7.10), we used the relation (see [86], 5.7.33, 4, p. 690) ∞  n=1

μn J0 (μ n x) J0 (ax) = , (μ2n − a2 ) J1 (μ n ) 2J1 (a)

which we apply for a = iλR, x = r/R. We will see later that when we deal with time-independent solutions, the survival and splitting probabilities can be evaluated much simpler using the reciprocity theorem.

7.3 The reciprocity theorem for particle collection in the general case of Robin boundary conditions In this section, we prove a reciprocity theorem, which we use in the constructions of different stochastic algorithms. Let Γ be a surface enclosing a volume where a diffusional particle collection occurs. The surface is composed of parts Γ i (i = 1, . . . , N).

136 | 7 Splitting and survival probabilities We consider the basic case of a unit source of particles at r . The distribution of the diffusing particles is governed by the diffusion equation D∆G(r, r ) − τ−1 G(r, r ) = −δ(r − r ),

(7.11)

where D is the diffusion coefficient and τ is the particle’s lifetime. We first present the reciprocity theorem given in [98], which concerns the case when on the boundary Γ 1 where the particles flux  ∂G(ξ , r) dσ(ξ ) (7.12) I(r) = −D ∂ν ξ Γ1

is calculated, the vanishing boundary conditions are prescribed: G(r, r ) = 0 for r ∈ Γ1 ,

(7.13)

while on the surfaces Γ2 , . . . , Γ N , the Robin boundary conditions are posed: −

∂G(r, r ) = S i G(r, r ) ∂ν i

for r ∈ Γ i (i = 1, 2, . . . , N).

(7.14)

Here, S i are reduced (i.e., divided by D), surface recombination velocities (0 ≤ S i ≤ ∞). The reciprocity theorem says that I(r) ≡ F(r) where F(r) is the unique solution of the following homogeneous BVP: D∆F(r) − τ−1 F(r) = 0,

(7.15)

with the modified boundary condition on Γ1 : F(r) = 1 for r ∈ Γ1

(7.16)

but with the same boundary conditions on other surfaces: −

∂F(r) = S i F(r) ∂ν i

for r ∈ Γ i (i = 2, . . . , N).

The proof immediately follows from the Green formula  ∂G ∂F ! F(r ) = −D F −G dσ. ∂ν ∂ν

(7.17)

(7.18)

Γ

Indeed, the integral over Γ is a sum of the integral over Γ1 and the integrals over Γ i , i = 2, . . . , N. The last surface integrals over Γ i , i = 2, . . . , N all vanish since G and F satisfy the same boundary conditions. In the surface integral over Γ1 , we have, due to

7.3 The reciprocity theorem

the boundary conditions, G ≡ 0 and F ≡ 1. Hence,  ∂G F(r ) = −D dσ, ∂ν1

| 137

(7.19)

Γ1

which completes the proof. Now we give the following generalization of this reciprocity theorem. Reciprocity Theorem. Assume that on all boundaries, including Γ1 , the Robin boundary conditions are prescribed: −

∂G(r, r ) = S i G(r, r ) ∂ν i

for r ∈ Γ i (i = 1, 2, . . . , N).

The flux of particles to any boundary, say, Γ k (k = 1, 2 . . .), defined as  ∂G(ξ , r) I(r) = −D dσ ∂ν k

(7.20)

(7.21)

Γk

satisfies the equality I(r) ≡ F(r) where F(r) is the unique solution of the following homogeneous BVP: D∆F(r) − τ−1 F(r) = 0,

(7.22)

with the modified boundary condition −

∂F(r) = S k (F(r) − 1) ∂ν k

for

r ∈ Γk

(7.23)

but with the same boundary conditions on other surfaces: −

∂F(r) = S i F(r) ∂ν i

for r ∈ Γ i , i  = k.

(7.24)

Proof. The proof follows by substitution of equality (7.23) in Green formula (7.18). Note that the case of the Dirichlet boundary conditions is a special case of this statement when S1 = ∞. This theorem can be used in many practically interesting cases when there is a need to calculate the particles fluxes on some specific parts of the boundary. In what follows, we show some practically interesting applications of this theorem. Let us start with a 1D case, which is simple. However, by using the splitting probabilities, quite a difficult problem of the nanowire growth simulation can be solved in the framework of this model.

138 | 7 Splitting and survival probabilities

7.4 Splitting and survival probabilities 7.4.1 Splitting probabilities for a finite interval and nanowire growth simulation The epitaxial growth of nanowires is mainly controlled by the diffusion of Ga and N atoms over the surface of nanowires. In [100], we suggested a stochastic model of nanowire growth that includes some ad hoc parameters. Here, we suggest a direct simulation of the surface diffusion, and this makes the model self-consistent. The technique presented provides an independent algorithm for simulation of the growth of nanowires. Let us find the splitting probabilities for a finite interval [0, R]. The governing diffusion equation reads 1 n(x ) = −δ(x − x), τ

D∆n(x ) −

x , x ∈ (0, L),

(7.25)

which is solved, generally, subject to the Robin boundary conditions: ∂n + S1 n = 0, ∂x ∂n D + S2 n = 0, ∂x

−D

for

x = 0,

for

x = R. 2

d In (7.25), τ is the mean diffusion time, and ∆ stands for the 1D Laplace operator, ∆ = dx 2. To simplify the notations, throughout the chapter, we use the ∆-operator notation in any dimension and in any coordinate system. According to the reciprocity theorem, the flux of particles to one of the boundaries (in our case, to one of the ends of the interval) which equals the splitting probability P1 defined as the probability for a particle starting at x to hit the origin x = 0 before it reaches the right end of the interval x = R, satisfies the following BVP:

∆n(x) − λ2 n(x) = 0, −

S ∂n S1 + n= 1, ∂x D D ∂n S2 + n = 0, ∂x D

x ∈ (0, R),

(7.26)

for

x = 0,

(7.27)

for

x = R,

(7.28)

 where λ = 1/Dτ is the inverse diffusion length. Let us first consider the case of no desorption, i.e., τ = ∞. The general solution is n = a + bx where the constants a and b are determined from the boundary conditions. This gives the explicit expression for the splitting probability P1 : P1 =

S1 D + S1 S2 (R − x) . D(S1 + S2 ) + S1 S2 R

7.4 Splitting and survival probabilities

|

139

In the case when the particle is absorbed on both ends of the interval, i.e., when S1 = ∞, S2 = ∞, the probability P1 has the following very simple form: P1 = 1 − x/R. The probability to first reach the right end x = R is thus P2 = x/R. In the general case when τ is finite, i.e., λ > 0, the general solution has the form n(x) = aeλx + be−λx . The constants a and b are found from boundary conditions (7.27), (7.28). From these simple equations, we find the solution n(x) =

! ( S2 − λ)eλx − e−λx e2λR ( SD2 + λ) S , (S 1 ) S D S D D2 − λ ( D1 − λ)( D2 − λ) − e2λR ( SD2 + λ)( SD1 + λ)

(7.29)

which is the probability P1 that a particle starting at a point x ∈ (0, R) first hits the left end x = 0 before it reaches the right end x = R or is absorbed inside the interval. One important example is the case of absorption on both ends of the interval, i.e., when S1 = S2 = ∞. From general formula (7.29), we obtain the splitting probabilities in the form P left =

sinh[λ(R − x)] , sinh[λR]

(7.30)

the probability to reach the left end x = 0, and P right =

sinh[λx] , sinh[λR]

(7.31)

the probability to reach the right end x = R. In the case of a half-line, i.e., for R = ∞, we should take the solution in the form n(x) = be−λx to satisfy the condition n(∞) = 0. The survival probability in this case is obtained from the boundary condition at x = 0 and takes the form n(x) =

1 e− λ x , 1 + λD S1

(7.32)

where x is the distance from the starting position to the origin x = 0. Using the explicit representations for the splitting and survival probabilities, the simulation of a particle motion on the surface of a nanowire (considered as an interval) is simple. The diffusion on the nanowire surface of the incoming flux of Ga and N atoms is simulated as follows. A particle launched on the side surface of a nanowire at a point x 1 ∈ (0, R) with the probability P left hits the bottom of the nanowire, and with the probability P right , it reaches the top of the nanowire where it makes the contribution to the nanowire growth. With probability P desorb = 1 − P left − P right , the particle is desorbed and is randomly (with isotropic direction) scattered and hits one of the neighbouring nanowires at some other height at a point x2 ∈ (0, R). The trajectory is simulated till the particle reaches the top of a nanowire, or it flies out of the simulation region.

140 | 7 Splitting and survival probabilities

This technique avoids simulation of the diffusion trajectories of a particle and is therefore extremely fast.

7.4.2 Survival probability for a disc and the exterior of a circular cylinder Let us start with the diffusion problem for the interior of a disk of radius R D∆n(r ) −

1  n(r ) = δ(r − r) τ

with the boundary condition D

∂n (R) + Sn(R) = 0. ∂ν

The reciprocity theorem says that the survival probability satisfies the same equation but with zero source, and in the right-hand side of the boundary conditions, the zero term should be replaced with S/D. The general solution of the homogeneous diffusion equation, which is rotationally symmetric, has the form n(r) = AI 0 (r/L) + BK0 (r/L), √

where we denote the diffusion length by L = τD, and I0 , K0 are the modified Bessel functions of the second kind. At r = 0, the solution must be finite, so B = 0, and the boundary condition reads A

! S 1 S I1 (R/L) + I0 (R/L) = , L D D

which implies that the survival probability is given by n=

I0 (r/L) . I0 (R/L) + SDL I1 (R/L)

Consider now an infinite circular cylinder of radius R and the axis coinciding with the axis OZ of the cylindrical coordinate system. The diffusion of particles starting at r in the exterior of the cylinder is governed by the diffusion equation D∆n(r ) −

1  n(r ) = δ(r − r) τ

with the boundary condition D

∂n (R) + Sn(R) = 0. ∂ν

7.4 Splitting and survival probabilities |

141

In this case, we should satisfy the conditions at the infinity. Therefore, in the general solution u(r) = AI0 (r/L) + BK0 (r/L), we should take A = 0, since I0 (∞) = ∞. Again, using the reciprocity theorem, we write down the Robin boundary conditions BμK1 (μR) + BλK0 (μR) = λ, where λ = S/D, μ = 1/L. Thus, the survival probability in the exterior of the cylinder has the form SProb =

K0 (r/L) , K0 (R/L) + SDL K1 (R/L)

(7.33)

where r = |r|. Therefore, the part of particles absorbed in the exterior of the cylinder reads I(r) = 1 − SProb. Note that for small values of r (say, r < L/4), the following approximation can be used: SProb =

ln(r/2L) + γ , ln(R/2L) + γ − D/(S · R)

where γ = 0.5772 . . . is the Euler constant. The exact expression for the survival probability outside of an infinite cylinder will be used in the next section to validate the random walk algorithm for the cathodoluminescence imaging technique.

7.4.3 Splitting probabilities for concentric spheres and annulus So far we have considered linear problems. Nonlinear problems where the diffusion particles are interacting with each other can also be solved by stochastic simulation algorithms, but probabilistic interpretations are not so easy to find. In the literature, the most known case concerns the homogeneous Smoluchowski coagulation equation where the pairwise interactions are simulated by the kinetic Monte Carlo (KMC) method. Inhomogeneous coagulation equations with diffusion were solved by some approximative models, which are very time consuming. In this chapter we suggest a very simple stochastic algorithm for solving this kind of nonlinear equations with diffusion, based on splitting probabilities for concentric spheres (3D) and annulus (2D). Let us consider the following problem. A diffusing particle starts at a point r in an annulus G with the inner radius R1 and outer radius R2 . Thus, R1 < r < R2 , where r = |r|. We are interested in splitting probabilities, i.e., in the probability P1 that the particle reaches first the inner boundary without hitting the outer boundary, and in P2 = 1− P1 , the probability that the diffusing particle reaches first the outer boundary without

142 | 7 Splitting and survival probabilities

hitting the inner boundary. No absorption inside the annulus is assumed. We assume general Robin boundary conditions. By the reciprocity theorem, the splitting probability, say, P1 = F(r), which is the flux to the inner boundary, satisfies the following BVP: D∆F(r) = 0, ∂F(r) + S1 F(r) = S1 , ∂n ∂F(r) D + S2 F(r) = 0, ∂n D

r ∈ G, for r = R1 , for r = R 2 ,

(7.34)

where ∆ is the radial Laplacian in 2D and n is the exterior unit normal vector to the boundary of the annulus. Solution of this problem can be found explicitly. Indeed, the general solution that depends only on the radial variable r has the form F(r) = A + B ln(r), where the constants are defined from the boundary conditions. These conditions read −

S B S1 + [A + B ln(R1 )] = 1 , R1 D D B S2 + [A + B ln(R2 )] = 0. R2 D

From these equations, we find the constants A and B, and the final expression for the probability P1 = F(r) is written in the form D R2 + r S2 R2 F(r) = . R D D ln 2 + + R1 S1 R1 S2 R2 ln

(7.35)

Note that in the case of pure absorption on the inner and outer boundaries (S1 = ∞, S2 = ∞), this expression simplifies to R2 r , F(r) = R2 ln R1 ln

(7.36)

which coincides with the known result given in Redner’s book [90]. Generalization to the 3D case, for a domain that is bounded by the inner sphere of radius R1 and outer sphere of radius R2 , is straightforward, keeping in mind that the general spherically symmetric solution of the Laplace equation has the form F(r) = A + B/r. From the boundary conditions, we find the constants A and B, which

7.4 Splitting and survival probabilities

|

143

gives the explicit expression:

R

 D −1 + r S2 R2 F(r) = .  R2 S  R2 D 1 + 22 2 −1 + R1 S2 R2 R1 S1 2

(7.37)

From this, in the case of pure absorptions on the boundaries, we find R2 −1 , F(r) = r R2 −1 R1

(7.38)

which coincides with the result presented in [90]. In a more general case, when the diffusion equation for an annulus involves an absorption inside, written as D∆F − 1τ F = 0, the general solution has the form u(r) = AI0 (r/L) + BK0 (r/L). Then, in contrast to the case of a disc, both functions I0 and K0 are present, and the Robin boundary conditions on the inner and outer boundaries read S1 , D AF3 (R2 ) + BF4 (R2 ) = 0,

AF1 (R1 ) + BF2 (R1 ) =

where     1 R1 R1 S1 − I1 , I0 D L L L     1 R2 R2 S F3 (R2 ) = 2 I0 + I1 , D L L L

F1 (R1 ) =

    1 R1 R1 S1 + K1 , K0 D L L L     1 R2 R2 S F4 (R2 ) = 2 K0 − K1 . D L L L

F2 (R1 ) =

From these equations, we find the constants A and B; thus, the solution is S P1 = 1 D

F4 (R2 ) I0 (r/L) F3 (R2 ) . F (R ) F2 (R1 ) − 4 2 F1 (R1 ) F3 (R2 )

K0 (r/L) −

Generalization to the case of a spherical shell between the outer sphere of radius R2 and inner radius R1 , with general Robin boundary conditions, can be derived using the general solution n(r) = a

cosh(λr) sinh(λr) +b . r r

(7.39)

144 | 7 Splitting and survival probabilities

From the boundary conditions, the constants a and b are obtained, and we present the final results: S1  sinh λr F21 (R2 ) cosh λr  − D r r F22 (R2 ) , (7.40) n(r) = F21 (R2 ) F11 (R1 ) − F12 (R1 ) F22 (R2 ) where sinh λR1 S1 sinh λR1 λ cosh λR1 + − , D R1 R1 R21 cosh λR1 S1 cosh λR1 λ sinh λR1 F12 (R1 ) = + − , D R1 R1 R21 λ cosh λR2 S2 sinh λR2 sinh λR2 F21 (R2 ) = + − , R2 D R2 R22

F11 (R1 ) =

F22 (R2 ) =

λ sinh λR2 S2 cosh λR2 cosh λR2 + − . R2 D R2 R22

The case for the exterior of a sphere, when R2 → ∞, is treated the same way, but we have to take b = 0 in general solution (7.39). The solution that is the survival probability for a particle starting at a distance h = r − R from the surface of the sphere of radius R has the form n(r) =

R r

e−λh . λD D 1+ + S SR

(7.41)

Remarkable, this result converges, as R → ∞, to the relevant 1D result for a half-line given by formula (7.32). This confirms that the survival probability of a 3D diffusion process for a half-space is defined by its 1D z-component only since R/(R + h) → 1 as R → ∞. In the problem of the cathodoluminescence, we will need the survival probability for a particle inside a sphere. The diffusion equation reads ∆n − λ2 n = 0 with the absorption boundary condition n(R) = 1 as follows from the reciprocity theorem. The general spherical symmetric solution of this diffusion equation is given by (7.39). In our case, b = 0, since the solution is finite at the origin. The boundary conditions yield R ; thus, we conclude that the survival probability is given by a = sinh(λR) P surv = n(r) =

sinh(λr) R . r sinh(λR)

(7.42)

In the cathodoluminescence problem, we will use the probability P absorpt = 1 − P surv , which is the probability that the particle starting inside the sphere at the distance r from the centre will be absorbed inside the sphere before it hits the surface of the sphere.

7.5 Cathodoluminescence |

145

We will need also the survival probability for a hemisphere with the same absorption boundary conditions on the spherical surface, and with the reflection (Neumann) boundary conditions on the plane part of the boundary. Due to the symmetry of the Wiener process with respect to the plane part of the boundary, it is clear that the survival probability has the same form given by (7.42).

7.5 Cathodoluminescence The cathodoluminescence imaging is a relatively new technique, which is used for imaging of microstructures like the threading dislocations in semiconductors (e.g. see [77, 82]), studies of sedimentary rocks [6] and many others. In this section, we apply the presented stochastic algorithm to the imaging of dislocations, which are considered as circular half-cylinders in a half-space, with the boundary z = 0. So let us consider an unbounded domain G, which is defined as the half-cylinder’s exterior overlapped with the half-space z < 0, see the general scheme presented in Fig. 7.1. We denote the plane z = 0 by Γ1 and the cylinder’s boundary by Γ2 . It is assumed that the cylinder axis coincides with the coordinate axis z.

crystal’s surface

Dislocation Source of excitons

Self-annihilation

Fig. 7.1. Random walk scheme for the cathodoluminescence imaging.

146 | 7 Splitting and survival probabilities

We assume that a source of excitons I(x) is placed in the domain G. The exciton is diffusing in the domain G with a constant diffusion coefficient D. When reaching the plane boundary z = 0, the exciton can be absorbed with a rate proportional to S1 . If not absorbed, it continues to diffuse in G. The same for the surface of the cylinder: when reaching Γ2 , the exciton is absorbed with a rate proportional to S2 . It means that both on the plane boundary and on the side surface of the cylinder, the Robin boundary conditions are prescribed. The exciton can be self-annihilated in the domain G, and the mean lifetime of an exciton before it annihilates in G is τ. The concentration density of excitons at a point x in G is governed by the following equation: 1 D∆n(x) − n(x) + I(x) = 0, τ

x∈G

(7.43)

and satisfies the boundary conditions: D

∂n(x) + S1 n(x) = 0, ∂ν

x ∈ Γ1 ,

(7.44)

D

∂n(x) + S2 n(x) = 0, ∂ν

x ∈ Γ2 .

(7.45)

Here, ν is the external unit normal vector at the point x on the boundary. It is convenient to introduce new variables l2D = τD, l S1 = τS1 and l S2 = τS2 , so that the diffusion length l D and l S1 , l S2 all have the length dimensionality. Equations (7.43)–(7.45) then read ∆n(x) −

1 n(x) + I(x)/D = 0, l2D ∂n(x) l S1 + 2 n(x) = 0, ∂ν lD ∂n(x) l S2 + 2 = 0, ∂ν lD

x ∈ G,

(7.46)

x ∈ Γ1 ,

(7.47)

x ∈ Γ2 .

(7.48)

We are interested in calculation of the total concentration of self-annihilated excitons in our domain G. To this end, we integrate equation (7.46) over the domain G:    1 ∆n(x)dx − 2 n(x) dx + I(x) dx = 0. (7.49) lD G

G

G

Using the Green formula and the boundary conditions, we find     l l 1 − S21 n(x)dx − S22 n(x)dx − 2 n(x) dx + I(x) dx = 0. lD lD lD Γ1

Γ2

G

G

(7.50)

7.5 Cathodoluminescence |

147

This shows that all the excitons originated from the source are distributed over three parts: the first is the set of excitons absorbed on the boundary Γ1 , the second are those absorbed on Γ2 and the third consists of excitons self-annihilated in G. Our goal is to calculate the concentration of self-annihilated excitons relative to the total concentration of excitons originated from the source.

7.5.1 The random WOS and hemispheres algorithm The random WOS algorithm [98] for solving the problem is based on the probabilistic interpretation of the diffusion process and the strict Markov property. The general idea behind the method is in tracking the excitons in our domain, and calculate the number of excitons that were self-annihilated in G. We can do it without introducing any grid. Indeed, let us consider a sphere with the centre placed at the exciton position generated by the source, and the radius r equal to the minimal distance to the boundary. It is known (e.g. see [98]) that the survival probability of the diffusion process inside this sphere is given by Pr = l D r/ sinh(l D r), while the random exit point from this sphere is uniformly distributed on the surface of the sphere. Based on this information, we can track the exciton trajectory: if the exciton survives in the first sphere, we consider the exit point as the centre of the second sphere. This process is then repeated until the exciton reaches the boundary or annihilates inside a sphere. When reaching the boundary, say, Γ1 , the exciton is absorbed with probability l Γ1 h/l2D , (1+l Γ1 h/l2D )

otherwise the exciton is reflected to the distance h along the normal vector

inside the domain G and proceeds to make the next diffusion step. Here, h is small enough. For the plane boundary with the reflection (Neumann) boundary conditions, the random walk simulation can be made more efficient. The case when on the plane boundary the reflection (Neumann) boundary conditions are imposed, the standard described random WOS algorithm can be improved. Indeed, in the standard implementation, the random walk may have long excursions near the boundary. To avoid multiple recurrence and long trajectories, we choose hemispheres instead of spheres, with the plane part coinciding with the plane boundary. So assume we are at a point whose distance to the plane boundary is less than to the cylinder’s surface. Then, we construct a hemisphere with this point lying on a radius that is perpendicular to the plane boundary. Now we have to find the survival probability of a Brownian motion inside this hemisphere, under the condition that on the plane boundary it is reflected, while on the spherical surface it is absorbed. Due to the symmetry of the reflecting Wiener process with respect to the plane, we conclude that the survival probability of the Brownian motion in the hemisphere with this boundary conditions is equivalent to the survival probability in a sphere with the same radius but with absorption boundary condition on the whole sphere. This survival probability is equal

148 | 7 Splitting and survival probabilities

to λR/ sinh(λR) provided the particle starts at the centre of the sphere. However, in our case, the particle inside the sphere has an arbitrary position. For this case, we have derived above the explicit expression for the survival probability (7.42). It remains to find the exit point of the reflected Brownian motion in the hemisphere. As explained, due to symmetry, we need to simulate the exit point of a particle starting at an arbitrary point in the hemisphere, and in the case the exit point x* is on the other side of the plane (z > 0), we just take a point inside the sphere that is symmetric to the point x* . Simulation of the exit point in a sphere that starts at an arbitrary point r inside the sphere can be efficiently simulated as follows. First, in the point r, an isotropically distributed unit vector is simulated, which defines a line along this direction. This line intersects the sphere in two points, say, point A and point B. Let the distance from r to A be d A and the distance from r to B be d B . Then, the exit point is taken as the point A with probability d B /(d A + d B ), and the exit point is taken as B with probability 1 − d B /(d A + d B ). To justify this algorithm, we mention that (1) a normal projection of the Wiener process inside the sphere on any line is a 1D Wiener process on this line, and (2) the exit probabilities of a Wiener process on a segment [A,B] starting at a point r ∈ [A, B] to the ends of the segment are given by the quantities d B /(d A + d B ) and d A /(d A + d B ). Let us prove these two properties. First, concerning the projection. Let us consider a 3D Wiener process w3 starting at a point x0 . The orthogonal projection of this process on any line that goes through this point is also a Wiener process on this line. Let us denote the unit vector that defines the line by n. The orthogonal projection of w3 on the line is defined as a linear transformation of w3 : w1 =

(w3 · n) n. (n · n)

Thus, w1 is obtained from w3 by this linear transform. Therefore, w1 is also Gaussian, it has also independent homogeneous increments and they both start from the same point. Note that a much more complicated proof of this simulation technique was given in [45]. Summarizing, let us describe the algorithm based on the random walk on spheres and hemispheres. Generally, the domain where we consider the transport of diffusing excitons can be considered as a half-space overlapped with a set of non-overlapping circular cylinders whose axes are perpendicular to the plane z = 0, the boundary of the domain. This boundary is denoted by Γ1 , and the set of all side boundaries of the cylinders is denoted by Γ2 . Thus, our domain has the plane boundary Γ1 and the boundary Γ2 . The governing equations and the Robin boundary conditions are given by (7.43–7.45). The algorithm can be described by the following steps.

7.5 Cathodoluminescence |

149

(1) Generate the starting position r0 according to the given source I(r). (2) Find the minimal distance, dmin , from r0 to the boundary of the domain. It may be the distance d Γ1 to the plane Γ1 or d Γ2 , the distance to the surface of one of the cylinders. (3) If d Γ1 < d Γ2 , i.e., the particle is closer to the plane boundary, we construct a hemisphere of radius d Γ2 (calculated as the distance to the closest cylinder) such that the particle lies on the radius that is perpendicular to the plane boundary. Calculate the survival probability Ps =

(4)

(5)

(6)

(7) (8)

sinh(λr0 )/λr0 , sinh(λR)/λR

where R = d Γ2 . With the probability 1 − Ps, the trajectory is terminated, and with probability Ps, we make the next step: simulate the exit point from the hemisphere. To this end, we simulate, as described above, the isotropic direction at the point x0 and find the points of intersection of the line having this direction with the sphere that is obtained by the symmetrical extension of the hemisphere relative to the plane boundary. If the simulated point x1 belongs to the hemisphere, it implies that we have found the exit point, so we put x0 = x1 and go to step 2. If the point is outside of the domain (i.e., it is lying on the reflected hemisphere), we just reflect this point symmetrically with respect to the plane and thus get the point x1 . Next, we put x0 = x1 and go to step 2. If d Γ1 > d Γ2 in step 3, then we construct the sphere of radius d Γ2 and calculate the survival probability Ps = λd Γ2 /sinh(λd Γ2 ). Now, with the probability 1 − Ps, the trajectory is finished, and with probability Ps, we sample the exit point on the sphere that is a point uniformly distributed on this sphere. If on some step, say, k, the walking particle is so close to the boundary Γ2 that the distance from the point to the surface Γ2 is less than a fixed small value ε, then the trajectory stops, if we deal with the absorption case (S2 = ∞). If two boundary events are possible, absorption or reflection, we proceed with sampling √ as described above at the beginning of section 7.5.1, taking the jump step h ∼ ε. The desired signal is calculated as F = M/N, where M is the number of trajectories absorbed inside the domain and N is the total number of simulated trajectories. The steps 1–7 are repeated for a set of starting points {x k }, and the desired signal is evaluated as F(x k ), a function of the distance from the dislocation to the point source.

It should be noted that the simulated trajectory of excitons always tends to reach the boundary. Strictly speaking, the Markov process of randomly walking excitons converges to the boundary; however, it reaches the boundary in infinite number of steps. To make the number of steps finite, one introduces an ε-boundary (ε-strip) in the neighbourhood of the boundary Γ2 , as mentioned implicitly in the description of

150 | 7 Splitting and survival probabilities

the algorithm in Step 6. This makes the number of steps in the random walk process finite, with a remarkably small mean value: N  ∼ | log(ε)|. In simulations, we have taken ε = 10−4 . The error due to the introduction of ε-boundary is of order of ε, so it is negligible compared to the experimental errors. We are interested in calculation of the function F(x), the total number of self-annihilated excitons in G, divided by the total number of excitons originated from the source, placed at the distance x from Γ2 , the cylinder’s boundary. To validate the algorithm, we compared the simulations with the exact result given by formula (7.33) for the case when the domain is the exterior of an infinite cylinder, and obtained a perfect agreement. We show now the simulation results for a more general case, for the domain composed of one semi-infinite circular cylinder and the half-space described above. In this case, the contrast curve can be well approximated analytically using the probabilistic interpretation and the survival probability. The 3D diffusion can be considered as a superposition of the radial diffusion and independent 1D diffusion along the axis OZ. The survival probability of the radial component is given by (7.33), and the 1D diffusion can be treated as a diffusion on the ray [0, −∞). The survival probability of a particle starting at a point −z with the Robin boundary condition with the surface recombination coefficient S1 was derived above (see (7.32) and (7.41)); it reads as P1 =

1 e−|z|/L . 1 + SD1 L

In a small neighbourhood around the circle, which is the intersection of the cylinder and the half-space, the 1D and the radial diffusions interact strongly, which however cannot affect much the integrals of the solution in the whole domain. To get closer, let us write down the solution n in the form n(r, z) = n1 (z) + n2 (r), where both n1 (z) and n2 (r) solve the same diffusion equation. These two solutions are coupled by the boundary conditions. The function n2 (r) is the solution in the exterior of the cylinder, and the exact expression for this function is given by formula (7.33). For the function n1 (z), we come to the problem D

d2 n1 (z) + S1 n1 (z) = 0, dz2 S dn1 S1 + (n1 + n2 ) = 1 , dz D D

z ∈ Γ1

−z/L 2 since dn , in the boundary dz = 0. We use the general form of solution, n 1 = ae conditions and find the constant a; thus,

P1 = n1 =

1 − u2 (r) −z/L e . 1 + SD1 L

Note that we have made only the first-order iteration, assuming that on the boundary z = 0, we use the solution n2 (r) for the exterior of the infinite cylinder. In the second

7.5 Cathodoluminescence |

151

iteration, we should take into account the value n1 (z) on the boundary r = R, etc. This leads to a correction factor β in the formula for P1 , which we have analyzed numerically P1 =

1 − u2 (r) −z/L e . 1 + βSD1 L

Note that the factor β is close to the value β = 8, and it slowly varies with the change of the diffusion length L. Therefore, using expression (7.33) for the function n2 (r), we conclude that for the surface recombination rate equal to S2 on the cylinder’s side and equal to S1 on the plane z = 0, the concentration of absorbed excitons in the whole domain is given by the following approximation: I = 1 − n1 − n2 = 1 −

! 1 K0 (r/L) , D 1 K0 (R/L) + S2 L K1 (R/L) 1 + β LS D

(7.51)

where r is the radial coordinate of the starting point and h = z, where −z is its z-coordinate; we have assumed here that |z| R0 , we apply the algorithm used in the case of the spherical molecule and based on the randomization of the Poisson integral (for |x| > R0 ) u(x) =

1 4πR0



|x |2 − R 20 u(y)dσ(y). |x − y |3

(9.5)

S(0,R0 )

So, with probability 1 − R/R0 , the particle moves to infinity (x1 = ∞), N = 1 and ξ (x0 ) = 0. With the complementary probability, R/R0 , the particle is placed on the sphere S(0, R0 ). In this case, there are two possibilities. The first is that |x1 − x c | > R c . This means that x1 lies on the boundary of the molecule ∂G. Here, we have to utilize the boundary conditions. Suppose now that |x1 − x c | < R c . In this case, we can use the Poisson kernel for the interior of a ball B(x c , R c ), which differs from (9.5) only in its sign. So, the next point, x2 , is chosen randomly on S(x c , R c ) in accordance with the Poisson’s density 2 2 1 R c −|x1 | 4πR c |x1 −y|3 . Part of S(x c , R c ) coincides with the boundary of the molecule, where we apply the boundary condition. The rest of the surface of sphere S(x c , R c ) lies outside of B(0, R0 ), where we can once more apply formula (9.5) to simulate the next point, x2 , on the surface of the large sphere. Finally, the constructed Markov chain stops either at the absorbing boundary, where we set ξ (x 0 ) = 1, or at the infinity, where the estimate assumes a value of zero. Clearly, for a fixed x0 , the estimate ξ (x0 ) follows the binomial distribution. Apparently, the same reasoning makes it possible to use this efficient algorithm to construct a Monte Carlo estimate in the case when the molecule, G, can be represented as a sphere with several intersecting spherical cavities. Note that the described algorithm can also be applied to problems with stochastic gating of a cavity [123].

9.1.2 Capacitance calculations It is a common knowledge that there exists a deep analogy between mathematical models used in diffusion, heat conduction and electrostatics. This allows using the same approach to constructing computational algorithms for solving problems coming from different areas of physics. In particular, the same random walk simulations had been used for calculating hydrodynamic friction, diffusion limited reaction rate and electrical capacitance (see, e.g. [22, 30, 124]).

9.1 Diffusion-limited reaction rate

|

171

Here, we consider the problem of computing the electrostatic capacitance of an isolated body, G. It can be determined by calculation of the surface integral  1 ∂u dσ(y), C= − 4π ∂n ∂G

where u is the solution of the exterior Dirichlet problem for the Laplace equation with the boundary value equal to 1. Using the same reasoning as in Section 9.1.1, we reduce the problem to computation of the following estimate: R E ξ (x 0 ) = C,

(9.6)

where E(ξ (x0 )|x0 ) = u(x0 ) for a random point x0 uniformly distributed on the spherical surface, S(0, R), of a ball, which completely contains G. As an example, we consider the classical test problem in electrostatics: computation of capacitance for the unit cube. In Section 6.1, we showed how this problem can be solved by the walk-on-boundary algorithm. Here, we utilize the specifics of the problem’s geometry and apply the walk-in-subdomains technique (Section 9.2). In this particular case, the computational domain G e = R3 \ G can be represented as the unity of seven subdomains: six half-spaces G2i−1 = {x : x(i) < −1/2},

G2i = {x : x(i) > 1/2},

i = 1, 2, 3

(9.7)

and the exterior of the starting sphere S(0, R). Here, R must be greater or equal to √ 3/2. The algorithm of simulating a random walk and its exit points (to the boundary, ∂G, or to infinity) works as follows. It begins with the generation of an isotropically distributed random vector x0 of length R. We take this point as the starting point of a Markov chain x(0) = x0 . Suppose this chain was simulated up to the point x(i) . With probability 1, this point falls into one of the half-spaces (9.7). Let, for definiteness, x(i) ∈ G6 . For a half-space, the Green’s function of the Dirichlet problem, Φ(x, x(i) ), is well known, and the exit point for the Brownian motion in this domain is distributed in accordance with the density (i) 1 1 x(3) − x(3) z ∂Φ = . = ∂n(x) 2π |x − x(i) |3 2π (z2 + r2 )3/2

| the distance from x (i) to the plane Here, n = (0, 0, −1) and we denote by z = |x(3) − x(i) (3) / {x (3) = 1/2}, and r = (x (1) − x (i) )2 + (x(2) − x(i) )2 . (1) (2) Then, we turn to the polar coordinates and simulate the next point in the Markov + r cos φ, x(i) + r sin φ, 1/2) using the exact formulas φ = 2πα0 , chain, x(i+1) = (x(i) (1) (2)

172 | 9 Macromolecules properties / r = z α−2 1 − 1. Here, α are independent uniformly distributed in [0, 1] standard random variables. At every step of simulation, there is a non-zero probability that x(i) falls into the seventh subdomain, i.e., |x(i) | > R. For the exterior of a sphere, the Green’s function is exactly known, and we can use the same Poisson formula (9.5) as in the reaction-rate computation. Integration of the Poisson kernel gives 1 − R/|x(i) |. This means that with probability R/|x(i) |, the estimate is set to zero, while with the complement probability, the next point |x(i+1) | is simulated on S(0, R). Setting estimate to zero may be interpreted as the jump of the Markov chain to infinity. Note that E(ξ (x0 )|x0 ) is equal to probability of a Markov chain initiated at x0 to terminate on the surface of G. This means the second moment of E(ξ (x0 )|x0 ) equals its expectation, and the variance of ξ (x0 ) is bounded.

9.2 Walk in subdomains and efficient simulation of the Brownian motion exit points Consider the problem of efficient simulation of exit points from a domain G for the Brownian random process. For simple domains, it is possible to construct an exact representation of the Green’s function for the Dirichlet problem, Φ, and simulate exit points on the domain boundary Γ = ∂G in accordance with the normal derivative of this function, ∂Φ ∂n . Note, however, that even if Φ is constructed, the simulation algorithm could be prohibitively complicated and time-consuming. Commonly, the most reasonable solution in such situation is to apply the WOS algorithm (or analogous methods based on simulation of exit points from some other simple domains completely contained in G), which is stopped either exactly on Γ or in the ε-strip near the boundary. On the other hand, every step of constructing a random walk inside a domain requires the exact value of the distance from a current point to the boundary, Γ. Calculation of this value is the classical problem of computational geometry, and various approximate nearest neighbour algorithms can be efficiently applied for solving this problem in the case of a complicated surface [2, 3, 15]. One of the possible ways to overcome this algorithmic complexity is based on the representation of the whole domain in the form of a unity of intersecting subdomains [106]. We suppose that in every subdomain, simulation of a Monte Carlo estimate is simple and efficient. In a particular case, exact values of the Green’s functions could be used [73]. Consider the Dirichlet BVP for the Laplace equation (3.5) in a domain G=

M 2 j=1

Gj .

(9.8)

9.2 Walk in subdomains

|

173

Suppose that a) in every G j , there exists the unique solution u j of the Laplace equation with the Dirichlet boundary condition, g j ; b) for every u j , a Monte Carlo estimate can be constructed ξ [u j ](x) = g j (t* ) · Q(x; t* ; t1 , . . . , t N ),

(9.9)

where x ∈ G j , t* ∈ ∂G j and t i ∈ G j , i ≥ 1 is a Markov chain of length N. The weight Q does not depend on g j . Suppose that every subdomain G j , j = 1, . . . , M intersects with others:   G j ( k  = j G k )  = ∅. Next, its boundary can be represented as ∂G j = Γ j,1 Γ j,2 , where   Γ j,1 is a part of Γ, and Γ j,2 lies inside G. Thus, ∂G = j Γ j,1 , and Γ j,2 = k  = j ∂G j,k , where  ∂G j,k = ∂G j G k , i.e., the part of the subdomain G j boundary, which lies inside G k . Then, the estimate for the solution of (3.5) (9.8) can be constructed as follows: " K #  K k−1 k k k Q(t* ; t* ; t1 , . . . , t N k ) , (9.10) ζ [u](x) = g(t* ) k=0 0 0 where we set t−1 * ≡ x. Here, x ∈ G (0) and t 1 , . . . , t N0 is a random walk in G (0) such that 0 0 0 t* ∈ ∂G(0) . Then, either t* ∈ Γ(0),1 or t* ∈ Γ(0),2 . In the first case, the Markov chain simulation is terminated, while in the second case, t0* lies inside some other subdomain, which is denoted as G(1) . Finally, the process as a whole is terminated at some step, K. Further, we give the properties of the constructed estimate (see [106]). To prove them, we used an obvious extension of the Schwartz lemma (see, e.g. [12, 111])

Lemma 9.1. Let G be a union of two intersecting subdomains, G 1 and G2 . Suppose boundaries of these subdomains consist of a finite number of surfaces with continuously changing normal vector, and these boundaries, ∂G 1 and ∂G2 , intersect at angles separated from zero. Let F satisfy the Laplace equation in G1 . Suppose it is continuous in G1 , equal zero on Γ1,1 (i.e., on the boundary of G) and equal one on Γ1,2 (i.e., inside G2 ). Then, there exists a constant 0 < q < 1, which depends on G 1 and G2 only, such that F ≤ q on Γ2,2 (i.e., inside G1 ). Theorem 9.1. If ξ [u j ](x) defined by (9.9) is an unbiased estimate for a Dirichlet problem in every subdomain G j , then ζ [u](x) constructed in accordance with (9.10) is  the unbiased estimate for the Dirichlet problem in the whole domain, G = M j=1 G j . Theorem 9.2. Suppose estimates ξ [u j ](x) defined by (9.9) are biased: Eξ [u j ](x) = u j (x) + δ j (x), where |δ j (x)| ≤ εB j and B j = sup |u j (x)|. x∈G j

(9.11)

174 | 9 Macromolecules properties

Then, the estimate ζ (x) in the whole domain G is also biased ε | E ζ (x) − u(x)| ≤ B, 1 − (ε + q) where q = max q i,k , B = sup |u(x)|. i,k

x∈G

Note that for the WOS-based estimate, weights Q are less than 1 (see [23]). Therefore, the bias of ζ (x) coincides with the bias of ξ (x). Thus, it is less than Aω(ε) for some constant A, where ω is the continuity modulus of u(x). Theorem 9.3. Variance of ζ (x) is bounded.

9.3 Monte Carlo algorithms for boundary-value conditions containing the normal derivative Consider now the case of a general-type boundary condition (9.1). One of the possible ways of dealing with such condition could be utilizing the FD approximation to the normal derivative. Let y ∈ ∂G and x h,y = y + nh ∈ G1 , where h is a small number, and n = n(y) is the normal vector at y. Then, the finite-difference approximation of the boundary condition (9.1) can be written as u(y) = p h u(x h,y ) + (1 − p h ) + O(h2 ),

(9.12)

−1

where p h (y) = 1 + κ s (y)h . To take this condition into account via Monte Carlo, we D randomize it. When a diffusing particle hits the boundary at a point x i , we calculate p h (x i ). With probability 1 − p h , the particle is absorbed by the molecule and we set u(x i ) = 1. With the complementary probability, p h , we reflect the particle and set its position to x i+1 = x h,y . Every time a point of the Markov chain {x i , i = 0, 1, . . . , N } hits the boundary, equation (9.12) is used, which introduces an O(h2 ) bias. The resulting bias of the estimate for the reaction rate can be evaluated analytically in the simplest cases only.¹ Generally, the number of returns to the surface of a molecule is O(1/h), which means that the resulting bias is O(h). 9.3.1 WOS algorithm for mixed boundary-value conditions As an alternative to the FD approximation to flux conditions, we can use an exact integral formula for a function value at a point on the boundary [108, 109].

1 For example, let κ s = const, then the mean number of steps until the particle is absorbed is equal to (2D+3κ s h)(R0 +h) . Hence, the resulting bias is of order h. However, if κ s h  D, then it is O(h2 ). h(κ (R +h)+D) s

0

9.3 Monte Carlo for flux conditions |

175

Consider first the linearized Poisson-Boltzmann (PB) (Helmholtz) equation ∆u − κ2 u = 0, κ = const ≥ 0

(9.13)

in a bounded domain G ⊂ Rm , and a mixed BVP for it: α(y)

∂u (y) + β(y)u(y) = g(y), y ∈ Γ = ∂G. ∂n

(9.14)

Here, α = 1, β = 0 on Γ0 , and vice versa: α = 0, β = 1 on Γ1 = Γ \ Γ0 . We suppose the boundary is piecewise smooth and that the parameters of the problem guarantee the unique solution exists [72]. To simplify the inference and formulas involved, in the rest of the chapter, we consider only the 3D Euclidean space. It is clear, however, that the integral formula derived stays valid for an arbitrary m ≥ 2. To find the solution at a fixed point, x0 , we use the WOS algorithm. Let x ∈ G. We define d(x) as the distance from this point to the boundary Γ. Next, we consider the ball B(x, d(x)) and write down the integral Green’s formula for the solution, u, and the Green’s function for this ball, Φ κ,d :  ∂Φ κ,d u(x) = u ds. (9.15) ∂n S(x,d(x)) 1 sinh(κ(d−|y−x|)) Here, Φ κ,d (x, y) = − 4π , and S(x, d(x)) denotes the sphere of radius d(x) |y−x| sinh(κd) centred at the point x. Randomization of (9.15) leads to the estimate based on simulation of the WOS Markov chain. The chain is defined by the recursive relation: x i+1 = x i + d(x i ) ω i , where {ω0 , ω1 , . . .} is a sequence of independent isotropic unit vectors. From (9.15), we have u(x i ) = E(q(κ, d(x i ))u(x i+1 )|x i ), and we treat the factor,

q(κ, d(x i )) =

κd , sinh(κd)

(9.16)

as the survival probability. For κ = 0 (or if we use the weight estimate [21]), WOS with probability 1 converges to the boundary. Let x k be the first point of the Markov chain that hits Γ ε , the ε-strip near the boundary, and denote by x*k ∈ Γ the point nearest to x k . Then, u(x0 ) = E(u(x k )χ), where χ = 0, if the trajectory was terminated inside the domain, χ = 1, if the Markov $ chain reached Γ ε and χ = ki=0 q(κ, d(x i )), if we use the weight estimate. For x*k ∈ Γ1 , we have u(x k ) = g(x*k ) + ϕ1 (x k , x*k ), where ϕ1 (x k , x*k ) is O(ε) for elliptic x*k .

(9.17)

176 | 9 Macromolecules properties For x*k ∈ Γ0 , the boundary conditions give u(x k ) = u(x*k ) − g(x*k )d(x k ) + ϕ0 (x k , x*k ),

(9.18)

where ϕ0 (x k , x*k ) = O(ε2 ) as ε → 0. To be sure that the nearest point on the boundary, x*k , is elliptic, we use the walk-in-subdomains technique [106]. Note that for x*k ∈ Γ0 , the value of u(x*k ) is not known. To estimate it, we will use the integral formula derived in the next section.

9.3.2 Mean-value relation at a point on the boundary For an elliptic point on the boundary, x ∈ Γ0 , consider the ball B(x, a) and the Green’s function Φ κ,a for this ball taken at its centre. For every y  = x, we have ∆ y Φ κ,a − κ2 Φ κ,a = 0.  Denote by B i (x, a) = B(x, a) G the interior part of the ball that lies inside the domain, and let S i (x, a) be the interior part of its surface. We apply the Green’s formula to the pair of functions, u and Φ κ,a , in B i (x, a) \ B(x, ε), excluding a small neighbourhood of the point. Both functions satisfy the PB equation in this domain,  Φ κ,a = 0 on S(x, a), and we suppose that everywhere on Γ B(x, a), u satisfies the Neumann boundary conditions. Next, we take into account smoothness of u and that for elliptic points, the surface area of S i (x, ε) is 2πε2 (1 + O(ε)) when ε → 0. As a consequence, we obtain in the limit the following integral formula:  ∂Φ κ,a u(x) = 2 u ds ∂n ∂B i (x,a)\{x}



− Γ

2 Φ κ,a g ds. 

(9.19)

B(x,a)\{x}

Clearly, this formula stays

 valid when κ = 0. In this case, the Green’s function is 1 1 1 − Φ0,a (x, y) = − 4π a . |y−x| For convenience, we explicitly isolate singularities in the kernels of integral operators and rewrite (9.19) in the following form:  1 cos φ yx u(x) = Q κ,a u(y) ds(y) 2π |y − x|2 ∂B i (x,a)\{x}



+ Γ



B(x,a)\{x}

1 2π|y − x|

 1−

|y − x|

a

 Q1κ,a g(y) ds(y).

(9.20)

9.3 Monte Carlo for flux conditions

177

|

Here, cos φ yx is the angle between the external (with respect to B i (x, a)) normal vector at a point y and vector y − x. The weight function, Q κ,a (|y − x|) =

sinh(κ(a − |y − x|)) + κ|y − x| cosh(κ(a − |y − x|)) , sinh(κa)

is smooth, and Q κ,a = q(κ, a) =

κa sinh(κa)

on the surface of the auxiliary sphere, S(x, a).

Clearly, everywhere in B(x, a) \ {x}, this weight function is positive and its value is less than 1. For κ = 0, it equals 1 identically. a , is also smooth. It is less The second weight function, Q1κ,a = sinh(κ(a−|y−x|)) a−|y−x| sinh(κa) than or equal to 1, and Q1κ,a ≥

κa . sinh(κa)

Obviously, for κ = 0, it equals 1 identically.

9.3.3 Construction of the algorithm and its convergence Suppose first that the part of the domain’s boundary with the Neumann conditions, Γ0 , is convex. In this case, the kernel of the integral operator in (9.20) is sub-stochastic, which means that it is non-negative and its integral is less than or equal to 1. Therefore, we can use this kernel as the transition density. The term ∂B i (x*i , a)

cos φ x x* i+1 i 1 2π |x i+1 −x*i |2

corresponds to

the isotropic distribution of x i+1 ∈ in a solid angle with its vertex at x*i . The * weight, Q κ,a (|x i+1 − x i |), is treated as the survival probability on this transition. (Note that for the plane boundary, with probability 1, x i+1 is distributed isotropically on the half-sphere S i (x*i , a), and the survival probability equals q(κ, a)). The described construction of a Markov chain corresponds to the so-called direct simulation of an integral equation [23]. This means that the resulting estimate for the solution’s value at a point, x = x0 ∈ G, is ξ [u](x) =

N 

ξ [F](x i ).

(9.21)

i=0

Here, N is the random length of the Markov chain, and ξ [F] are estimates for the right-hand side of the integral equation. This function is defined by (9.17), (9.18), (9.20), which gives F(x) = 0, when x ∈ G \ Γ ε ;    |y − x* | 1 Q1κ,a g(y) ds(y) = 1− a 2π|y − x* |  Γ

B(x* ,a)\{x* }

−g(x* ) d(x) + ϕ0 (x, x* ), when x ∈ Γ ε and x* ∈ Γ0 ; = g(x* ) + ϕ1 (x, x* ), when x ∈ Γ ε and x* ∈ Γ1 .

(9.22)

For Markov chains based on the direct simulation, the finiteness of the mean number of steps, EN < ∞, is equivalent to convergence of the Neumann series for the

178 | 9 Macromolecules properties

correspondent integral operator, which kernel coincides with the transition density of this Markov chain. Besides that, the kernel of the integral operator that defines the second moment of the estimate is also equal to this density [23]. This means that for an exactly known free term, F, the estimate (9.21) is unbiased and has finite variance. The same is true if the estimates, ξ [F](x i ), are unbiased and have uniformly in x i bounded second moments. It is clear that we can easily choose such a density that the estimate for the integral in (9.22) will have the requested properties. To prove that the mean number of steps is finite, we consider the auxiliary BVP: ∆p0 − κ2 p0 = 0,

p0 |Γ1 = 0,

∂p0 | = 1. ∂n Γ0

(9.23)

For plane or spherical Γ0 , the integral in (9.22) equals (cosh(κa)− 1)/(κ sinh(κa)). 2 4 as κa → 0. For More generally, for elliptic points, it equals 2a 1 − (κa) 24 + O(κa) a/R → 0, we take ε = O(a/2R)3 , where R is the minimal curvature radius at the point ( ( a )) x*i ∈ Γ0 , and get F(x i ) = 2a 1 + O 2R . Consider the estimate (9.21) for p0 . Note that the mean-value relation (9.20) is  valid only when Γ B(x*i , a) ⊂ Γ0 . From here, it follows that radii of auxiliary spheres, a i , cannot be greater than the distance from x*i to the boundary between Γ1 and Γ0 and can be arbitrary small. We take it as the maximal possible and calculate (for κ = 0) the termination probability for such random walk. For small distances, it can be easily shown that the probability, q, of a reflected diffusion coming to Γ1 and thus being terminated is separated from zero. Therefore, the mean number of reflections near the line that separates Γ0 and Γ1 is bounded by some constant, c* /q. Next, we fix some value, a* , and divide the set of all reflections into two classes: in the first one, we put reflections with radii, a i , which are greater than a* , and the second class includes the reflections with a i < a* . Let N0,1 and N0,2 be, respectively, mean numbers of reflections in the correspondent classes. Then, we have p0 > N0,1 a* /2 and, hence, the overall number of reflections, N0 = N0,1 + N0,2 , is less than C a = 2p0 /a* + c* /q. From here, it follows that the overall mean number of steps of the described version of the WOS algorithm is of the same order as for the case of the Dirichlet boundary conditions, which means EN < const| log(ε)|. In the previous reasoning, we supposed the error functions ϕ0 and ϕ1 to be known exactly. This presupposition provided us with a possibility of proving that the estimate (9.21) is unbiased. To obtain a functioning estimate, we are forced to get rid of these unknowns. The resulting bias in the estimate is ϕ1 + N0 ϕ0 , which is O(ε) when ε → 0. This means that we proved the following proposition. Theorem 9.4. The new version of the WOS algorithm provides the estimate, ξ [u](x), for mixed BVP (9.13), (9.14). The variance of this estimate is bounded, and its bias is O(ε), where ε is the width of the strip near the boundary of the domain. For the required accuracy of solution, δ = ε, the computational cost of the algorithm is O(log(δ) δ−2 ). For κ > 0 and for pure Neumann BVP, the algorithm has the same properties.

9.4 Continuity BVP

|

179

Note that with the FD approximation for the solution’s normal derivative, every hit of the ε-strip near the boundary, Γ0 , introduces O(ε + h2 ) bias into the estimate (h is the step value in the normal derivative approximation) [57, 69]. The mean number of reflections in this case is O(h−1 ). It means that if ε ∼ h2 , then the resulting bias is O(h), and the overall mean number of steps is O(log(h) h−1 ). Thus, for the required accuracy δ, we have to take ε ∼ δ2 and h ∼ δ. The computational cost of this algorithm is O(log(δ) δ −3 ). So, as we see, the approach we described here makes it possible to substantially improve the efficiency of the WOS algorithm when applied to solving mixed and Neumann BVPs. The algorithm described above works for a convex Γ0 . If it is not the case, the kernel of the integral operator in (9.20) can be negative, and the direct simulation is not feasible. This also means that changing the kernel of the integral operator to its modulus can lead to non-convergent Neumann series, and, thus, to non-operational Monte Carlo estimates. To solve this problem, we propose using a simple approximation to the integral relation. Consider the tangent plane at the point, x*i ∈ Γ0 , and simulate the next point of the Markov chain isotropically on the half-sphere, S− (x*i , a), which lies inside the domain. It can be easily shown that for small a/R, such algorithm introduces an O(a/2R)3 error. Therefore, the resulting bias is O(a/2R)2 , and the computational cost of such biased algorithm is O(log(δ) δ−5/2 ).

9.4 Continuity BVP We consider the BVP for a function, u(x), that satisfies different elliptic equations with constant coefficients inside a bounded simple connected domain, G i ⊂ R3 , and in its exterior, G e = R3 \ G i . In Section 3.7, we considered such problem and constructed the walk-on-boundary algorithm for its solution. Here, we describe the alternative Monte Carlo method, which is based on simulation of the WOS Markov chain. Denote, for convenience, by u i (x) and u e (x) the restrictions of function u(x) to G i and G e , respectively. Let the first function satisfy the Poisson equation ϵ i ∆u i = −4πρ,

(9.24)

and the second one satisfy the linearized PB equation ϵ e ∆u e − ϵ e κ2 u e = 0,

(9.25)

where ϵ e ≥ ϵ i and κ could be equal to zero. The continuity conditions on the piecewise smooth boundary, Γ, relate limiting values of the solutions, and their fluxes as well: u i (y) = u e (y), ϵ i

∂u i ∂u e (y) = ϵ e (y), y ∈ Γ. ∂n ∂n

(9.26)

180 | 9 Macromolecules properties Here, the normal vector, n, is pointed out into G e , and u e (x) → 0 as |x| goes to infinity. We assume that the parameters of the problem guarantee the unique solution exists [72]. Problems of this kind arise in various scientific applications such as diffusion, heat conduction, geophysics, transport theory, etc., where the solution of an elliptic equation with piecewise constant coefficients should be found in a non-uniform computational domain. In the molecular biophysics applications [14], such problems arise when the electrostatic properties of macromolecules are needed to be computed. In this case, G i can be thought of as a molecule in aqueous solution. In the framework of the implicit solvent model, only the geometric structure of this molecule is described explicitly, whereas the surrounding water with ions dissolved is considered a continuous medium.

9.4.1 Monte Carlo method To solve numerically the problem (9.24), (9.25), (9.26), we propose to use a Monte Carlo method [109, 110]. There are several reasons for such a choice. FD and other deterministic computational methods that are commonly used for solving elliptic BVPs encounter apparent complications when applied to calculating electrostatic properties of molecules in solvent. Most of these difficulties are caused by the complexity of molecular surface. On the other hand, the most efficient and commonly used Monte Carlo methods such as the WOS algorithm [21, 23, 76] and the random walk-on-boundary algorithm [104] can analytically take the geometry into account. The latter can be applied to solving not only Dirichlet, Neumann and third BVPs but also for the problems with continuity boundary conditions in the case when κ = 0 [49, 50]. This method works well for small molecules but becomes computationally expensive for larger structures, which means that it needs substantial optimization and further algorithmic development. It is well known that the WOS algorithm is designed to work with the Dirichlet boundary conditions. With this method, the common way of treating flux conditions is to simulate reflection from the boundary in accordance with the FD approximation to the normal derivative [35, 57, 64, 69]. Such an approach has a drawback. It introduces a bias into the estimate and substantially elongates simulation of Markov chain trajectories. In [107], we proposed a new approach to constructing Monte Carlo algorithms for solving elliptic BVPs with flux conditions. This approach is based on the mean-value relation written for the solution value at a point, which lies exactly on the boundary. It provides a possibility to get rid of the bias when using Green’s function–based random walk methods and treating algorithmically boundary conditions that involve normal derivatives. Consider the problem of computing the solution to (9.24), (9.25), (9.26) at a fixed point, x0 . Suppose for definiteness that x0 ∈ G i . To estimate u(x0 ), we represent it as

9.4 Continuity BVP |

181

a sum of the regular part and the volume potential: u(x0 ) = u0 (x0 ) + g(x0 ).  Here, g(x0 ) = G i ϵ1i |x01−y| ρ(y)dy. In the molecular electrostatic problems, charges are considered to be concentrated at a finite number of fixed points. Hence, 1 1 g(x0 ) = M m=1 ϵ i |x0 −x c,m | ρ m . Usually, in these problems, x 0 coincides with one of x c,m . The regular part of the solution satisfies the Laplace equation in G i . Therefore, we have a possibility to use the WOS algorithm to find it. Let x0 be the starting point and d(x i ) be the distance from the given point, x i , to the boundary, Γ. Generally, WOS Markov chain is defined by the recursive relation: x i+1 = x i + d(x i )ω i , where {ω0 , ω1 , . . . } is a sequence of independent isotropic unit vectors. With probability 1, this chain converges to the boundary [21]. Let x k be the first point of the Markov chain that hits Γ ε , the ε-strip near the boundary. Denote by x*k ∈ Γ the point nearest to x k . Clearly, the sequence {u0 (x i ), i = 0, 1, . . . } forms a martingale. Therefore, u0 (x0 ) = Eu0 (x k ) = E(u(x k ) − g(x k )) = E(u(x*k ) − g(x*k ) + ϕ), where ϕ = O(ε), for elliptic boundary points, x *k . The mean number of steps in the WOS Markov chain before it for the first time hits the ε-strip near the boundary is O(log ε) [21]. In the molecular electrostatics problems, however, it is natural to use the more efficient way of simulating exit points of Brownian motion on the boundary, Γ. The algorithm is based on the walk-in-subdomains approach [106]. Such construction utilizes the commonly used representation of a molecular structure in the form of a union of intersecting spheres (Fig. 9.2). This version of the WOS algorithm converges geometrically, and there is no bias in the estimate, since the last point of the Markov chain lies exactly on Γ.

Fig. 9.2. Example of a model geometry and walk in subdomains in the energy calculation.

182 | 9 Macromolecules properties Note that to compute the estimate for u0 (x0 ), we use the unknown boundary values of the solution. With the Monte Carlo calculations, we can use estimates instead of these values. In the next section, we derive the mean-value relation, which is then used in Section 9.4.3 to construct such an estimate.

9.4.2 Integral representation at a boundary point To elucidate the approach we use, we consider first the exterior Neumann problem for the PB equation (9.25) in G e : ∂u e (y) = f (y), y ∈ Γ. ∂n

(9.27)

Let x ∈ Γ be an elliptic point on the boundary. To construct an integral representation for the solution value at this point, consider the ball, B(x, a), of radius a, and x being its centre. 1 sinh(κ(a−|x−y|)) be the Green’s function of the Dirichlet problem Let Φ κ,a (x, y) = − 4π |x−y| sinh(κa) for PB equation (9.25) considered in the ball, B(x, a), and taken at the central point  of this ball. Denote by B e (x, a) = B(x, a) G e the external part of the ball, and let S e (x, a) be the part of the spherical surface that lies in G e . Next, we exclude from this ball a small vicinity of the point, x. From here, it follows that, for arbitrary ε < a, both functions, u e and Φ κ,a , satisfy the PB equation in B e (x, a) \ B(x, ε). Therefore, it is possible to use the Green’s formula for this pair of functions in this domain. Taking the limit of this formula as ε → 0, we have  ∂Φ κ,a u e ds u e (x) = 2 ∂n Se

 2

− Γ



B(x,a)\{x}



+ Γ

∂Φ κ,a u e ds ∂n

2Φ κ,a 

∂u e ds. ∂n

(9.28)

B(x,a)\{x}

Here, in the second and third integrals, we took into account that, on Γ, the normal vector external with respect to B e (x, a) \ B(x, ε) has the direction opposite to the normal vector we use in boundary conditions (9.27). The normal derivative of the Green’s function can be written down explicitly. We have ∂Φ∂nκ,a (y) = Q κ,a (r) ∂Φ∂n0,a (y). cosh(κ(a−r) < 1, r = |x − y|, Here, Q κ,a (r) = sinh(κ(a−r))+κr sinh(κa) cos ϕ

yx 1 and 2 ∂Φ∂n0,a (y) = 2π 2 , where ϕ yx is the angle between n(y) and y − x. |x−y|

1 1 1 Φ0,a (x, y) = − 4π |x−y| − a is the Green’s function for the Laplace equation.

9.4 Continuity BVP

|

183

Application of the Green’s formula to the pair of functions, u i and Φ κ,a , in B i (x, a) \ B(x, ε) provides the analogous result. To have a possibility to do this, we have to suppose that there are no charges in this part of the interior domain. From here, it follows that the total potential, u i , satisfies the Laplace equation. Thus, using the Green’s formula and taking the limit as ε → 0, we obtain  ∂Φ κ,a u i (x) = 2 u i ds ∂n Si



+

2 

Γ

∂Φ κ,a u i ds ∂n

B(x,a)\{x}



[−2κ2 Φ κ,a ]u i dy

+ B i (x,a)

 2Φ κ,a

− 

Γ

∂u i ds. ∂n

(9.29)

B(x,a)\{x}

Note the additional volume integral in this representation. To make use of continuity boundary conditions (9.26), we multiply (9.28) by ϵ e and (9.29) by ϵ i , respectively, and sum up the results. This gives us  ϵe 1 κa u(x) = u e ds ϵe + ϵi 2πa2 sinh(κa) S e (x,a)

+



ϵi ϵe + ϵi

1 κa u ds 2πa2 sinh(κa) i

S i (x,a)



ϵe − ϵi − ϵe + ϵi Γ

+



ϵi ϵe + ϵi

1 cos ϕ yx Q κ,a u ds 2π |x − y|2

B(x,a)\{x}



[−2κ2 Φ κ,a ]u i dy.

(9.30)

B i (x,a)

In the limiting case, κ = 0, this relation simplifies  ϵe 1 u(x) = u e ds ϵe + ϵi 2πa2 S e (x,a)



ϵi + ϵe + ϵi

1 u ds 2πa2 i

S i (x,a)



ϵe − ϵi − ϵe + ϵi Γ



1 cos ϕ yx u dy. 2π |x − y|2

B(x,a)\{x}

Note that for a plane boundary, cos ϕ yx = 0, and the integral over Γ vanishes.

(9.31)

184 | 9 Macromolecules properties

9.4.3 Estimate for the boundary value To construct an estimate for the solution of the BVP, we need an estimate for the unknown boundary value. To do this, we use the mean-value relation constructed in the previous section. It is not so simple as it may seem at the first sight, since we have to iterate integral operators standing in the right-hand sides of these representations. The kernels of these operators can be alternating, and the convergence of the Neumann series after replacing the kernel by its modulus cannot be ensured. As a consequence, the direct randomization of integral relations (9.28), (9.30), (9.31) cannot be used to constructing a Monte Carlo estimate [23, 24]. However, as we see, the only part of the integral operator, which can be negative, is the integral over the boundary. Thus, if Γ consists of planes, then the direct randomization is really possible. In the general case, we can use a simple probabilistic approximation. Let x = x*k be the point on the boundary nearest to the last point of the already constructed random walk. Consider first the exterior Neumann problem. If Γ is concave everywhere in B(x, a), then the kernel of the integral operator in (9.28) is positive, and the next point of the Markov chain, x k+1 , can be sampled isotropically in a solid angle with x being its vertex. However, it would be more natural to suppose that the boundary was convex. In this case, we draw the tangent plane to Γ at this point and sample x k+1 isotropically on the external hemisphere, S+ (x*k , a). Then, for sufficiently smooth function, u, we have Lemma 9.2.   u(x*k ) = E u(x k+1 ) | x*k + ϕ Γ , where ϕ Γ = O point, x *k .

(

) a 3 2R

as a/2R → 0. Here, R is the minimal radius of curvature at the

This statement is easily verified by expansion of u into series and direct integration. The same approach works in the case of continuity boundary conditions (9.26). For κ = 0, with probability p e = ϵ iϵ+ϵe e , we sample the next point of the Markov chain, x k+1 , isotropically on S+ (x*k , a). With the complementary probability, p i , the next point is chosen on the other hemisphere, S− (x*k , a). If κ > 0, with probability p e , we sample the next point on S+ (x*k , a) and treat the κa coefficient q(κ, a) = sinh(κa) as the survival probability. With probability p i , we sample the direction of the vector, x k+1 − x, isotropically pointing to S− (x*k , a). Next, with probability q(κ, a) (Q κ,a , if this vector intersects Γ), the next point is taken on the surface of the sphere, and with the complementary probability, it is sampled inside the ball, B− (x*k , a). The simulation density of r = x k+1 − x is taken to be consistent with sinh(κ(a − r)).

9.4 Continuity BVP

|

185

It is easy to prove that Lemma 9.2 is also valid for the proposed randomized approach to treating the continuity boundary conditions. It is essential to note that a randomization of the FD approximation for the normal derivative with a step, h, provides an O(h2 ) bias, both for the Neumann and the continuity boundary conditions.

9.4.4 Construction of the algorithm and its convergence To complete the construction, we need a Monte Carlo estimate for the solution value at an arbitrary point in the exterior domain, G e . It is natural to use the estimate based on simulation of the WOS Markov chain. Every step of the algorithm is the direct  u ds, randomization of the mean-value formula for the PB equation: u(x) = S(x,d) q(κ,d) 4πd2 where S(x, d) is the surface of a ball totally contained in G e and d is usually taken to be the maximum possible, i.e., equal to the distance from x to Γ. For κ = 0, we use the modification of the WOS algorithm with the direct simulation of jump to the absorbing state of the Markov chain at infinity [24]. The conditional mean number of steps for the chain to hit the ε-strip near the boundary is O(log ε). For positive κ, we consider q(κ, d) the survival probability. In this case, the WOS either comes to Γ or terminates at some finite absorbing state in G e . To prove the convergence of the algorithm, we reformulate the problem in terms of integral equations with generalized kernels [21, 23]. Inside the domains, G i and G e , we consider the mean-value formulas as such equations. In Γ ε , we use the approximation u(x) = u(x* ) + ϕ(x, x* ) and substitute the integral representation for u(x* ) at a boundary point, x* . Hence, the described random-walk construction corresponds to the so-called direct simulation of (approximated) integral equation [23]. This means that the resulting estimate for the solution’s value at a point, x = x0 ∈ R3 , is ξ [u](x) =

N 

ξ [F](x i ).

(9.32)

i=0

Here, N is the random length of the Markov chain, and ξ [F] are estimates for the right-hand side of the integral equation. For the external Neumann problem, this function is F(x) = 0, when x ∈ G e \ Γ ε ;  1 r 1 =− 1− Q (r) f (y) ds(y) 2πr a κ,a  Γ

B(x* ,a)\{x* }

+ϕ Γ + ϕ(x, x* ), when x ∈ Γ ε , where Q 1κ,a (r) =

sinh(κ(a−r)) a , a−r sinh(κa)

r = | y − x * |.

(9.33)

186 | 9 Macromolecules properties

For Markov chains based on the direct simulation, finiteness of the mean number of steps, EN < ∞, is equivalent to convergence of the Neumann series for the corresponding integral operator, which kernel coincides with the transition density of this Markov chain. Besides that, the kernel of the integral operator that defines the second moment of the estimate is also equal to this density [23]. This means that for the exactly known free term, F, estimate (9.32) is unbiased and has finite variance. The same is true if estimates, ξ [F](x i ), are unbiased and have uniformly in x i bounded second moments. It is clear that we can easily choose such a density that an estimate for the integral in (9.33) will have the requested properties. To prove that the mean number of steps is finite, we consider the auxiliary BVP: ∆p0 (x) − κ2 p0 (x) = 0, x ∈ G e ,

∂p0 | = −1. ∂n Γ

(9.34)

For this problem, the integral in (9.33) equals − 1)/(κ

(cosh(κa)  sinh(κa)) when Γ (κa)2 4 a is a plane or a sphere. This is equal to 2 1 − 24 + O(κa) , as κa → 0. Setting ε = O(a/2R)3 , we have Lemma 9.3. The mean number of boundary hits in the WOS solving the exterior ) ( Neumann problem is EN * = 2pa 0 1 + O(a2 ) . In the full analogy, we obtain the following. Lemma 9.4. The mean number of boundary hits of the WOS algorithm solving the continuity BVP is EN * = 2pa 1 (1 + O(a)). Here, p1 is a bounded solution to the problem (9.24), (9.25) with no charges in G i and with the boundary condition ϵ i ∂p∂n1,i (y) = ϵ e ∂p∂n1,e (y) + 1. Denote by {x*k,j ∈ Γ, j = 1, . . . , N i* } the sequence of exit points from G i for the WOS Markov chain used to calculate the solution of the continuity BVP. Let {x k+1,j ∈ G i , j = 1, . . . , N i* − 1} be the sequence of return points. Clearly, EN i* = p i E N * . Then, we have Theorem 9.5. The quantity g(x0 ) + ξ [u0 ](x0 ), where !

N i* −1

ξ [u

0

](x0 ) = −g(x*k,1 ) +



g(x k+1,j ) − g(x*k,j+1 )

(9.35)

j=1

is the estimate for the solution of the BVP (9.24), (9.25), (9.26). For ε = (a/2R)3 , the bias of this estimate is O(a)2 as a → 0. The variance of this estimate is finite, and the computational cost is O(log(δ) δ−5/2 ), for a given accuracy, δ. The variance is finite since the algorithm is based on the direct simulation of the transformed integral equation [23]. The logarithmic factor in the estimate comes from the mean number of steps in the WOS Markov chain until it hits for the first time the ε-strip near the boundary [21, 24].

9.5 Computing macromolecule energy

| 187

The same simulation can be used for computing the gradient of the solution ∇u(x0 ) = ∇g(x 0 ) + EQ0 ξ [u 0 ](x 0 ).

The vector weight Q0 is computed at the first step of the algorithm, when simulating the jump to the sphere S(x0 , d(x0 )) [21]. Let x0 ≡ (x0,(1) , x0,(2) , x0,(3) ). Then, 3(x1,(i) − x0,(i) ) ∂u0 (x) = E u0 (x1 ), i = 1, 2, 3, ∂x0,(i) d(x0 )2 i.e., Q0 = 3(x1 − x0 )/d(x0 )2 . It is essential to note that with the FD approximation of the normal derivative using the step h, the mean number of boundary hits is O(h−1 ). Therefore, the bias of the resulting estimate is O(h), and the computational cost is O(log(δ) δ−3 ) [57, 69]. Thus, even the simplest approximation to the (exact!) integral relation we constructed substantially improves efficiency of the WOS algorithm.

9.5 Computing macromolecule energy Biological macromolecules, such as peptides and proteins, contain charged and polar groups and exist in a heterogeneous cellular environment, which contains ions, water and other small molecules. Thus, it is not surprising that in vivo and in vitro studies show that salt-mediated long-range electrostatic interactions play a crucial and often complex role in the structure, stability, folding, dynamics and binding behaviour of biomolecules. For instance, it is well known that the thermodynamic binding affinity of charged ligands (e.g. metal ions, proteins and peptides) to proteins, the kinetic rates of protein–protein association and the stability of proteins can change significantly with small changes in salt concentration of the ionic solution [1, 16, 33, 61, 66, 80, 117]. To calculate salt-mediated electrostatic interactions, all-atom explicit solvent molecular dynamics simulations of the biomolecules in their natural environment [27, 84, 119] could be used. However, due to the extreme complexity of such physical model that should involve an enormous number of atoms, this approach is not feasible for routine calculations. On the other hand, the linear PB equation, which is based on an implicit solvent model but still treats the biomolecule at the molecular level of detail, has been successfully used to predict and interpret numerous salt-mediated electrostatic effects [4, 17, 20, 48]. Performing linear PB calculations of salt-dependent biomolecular properties can be problematic due to the inherent approximations of the deterministic PB numerical methods normally used. Such approach includes discretization errors and a rather crude treatment of the boundary conditions. Here, we describe an application of a grid-free Monte Carlo linear PB solver that can deliver non-specific salt-dependent electrostatic properties such as electrostatic solvation free energies with very high accuracy.

188 | 9 Macromolecules properties

9.5.1 Mathematical model and computational results Within the framework of the adopted here implicit solvent model, a polarizable solvent (aqueous salt solution) is considered as an infinite dielectric continuum, in which a cavity G i having the geometry of a molecule is immersed. Distribution of the electrostatic potential Φ(x) in the system satisfies the Poisson equation in the whole space (in CGS units): −∇ϵ ∇Φ(x) = 4πρ(x), x ∈ R3 .

(9.36)

Inside the molecule, point charges Q m , m = 1, . . . , M are located at the atomic centres, x(m) , respectively. This means the potential Φ i (x) satisfies ϵ i ∆Φ i (x) + 4π

M 

Q m δ(x − x(m) ) = 0, x ∈ G i ,

(9.37)

m=1

where ϵ i is constant dielectric permittivity. Outside the molecule, the charge distribution is determined by the mobile ions. In the equilibrium state, concentration of a particular kind of ions, j, is described by the Boltzmann law:   −z j eΦ(x) 0 . M j (x) = M j exp kB T Here, z j e is the charge of a solitary ion, e is the protonic charge, k B is the Boltzmann’s constant and T is the absolute temperature. In supposition that there is only 1:1 salt in the solution that dissociates into an equal number of ions of opposite sign, we get   z eΦ(x) , ρ(x) = −2ez1 M sinh 1 kB T where z1 = −z2 = 1, M10 = M20 ≡ M. Denote by ϵ e ≥ ϵ i the dielectric permittivity of the solution. Then,   z eΦ e (x) = 0, x ∈ G e , (9.38) ϵ e ∆Φ e (x) − 8πez1 M sinh 1 kB T Here, Φ e (x) is the potential outside the molecule, G e = R3 \ G i . Denote c j = 1000M 0j /N A the molar concentration, where N A is the Avogadro number, and introduce the ionic strength of solution (in M), 1 2 cj zj . 2 NI

Is =

j=1

In this particular case, N I = 2 and I s = 1000M/N A .

9.5 Computing macromolecule energy

u=

| 189

From here, we get that the dimensionless (unitless) electrostatic potential e k B T Φ satisfies the non-linear PB equation ∆u e (x) − κ2 sinh(u e (x)) = 0, x ∈ G e .

(9.39)

Here, κ2 =

8πN A e2 I s ϵ e 1000k B T

(9.40)

is the square of the Debye-Hückel parameter (or inverse length). For small salt concentration and values of the electrostatic potential, equation (9.39) may be linearized. In the linear case, we can get back to Φ e (x), which satisfies the following equation: ∆Φ e (x) − κ2 Φ e (x) = 0, x ∈ G e .

(9.41)

The system of equations (9.37), (9.41) must be complemented by the boundary conditions at infinity: Φ e (x) → 0 as |x| → ∞ and on the boundary between G i and G e . On that surface, the continuity conditions for both potential and flux must be satisfied [41, 113]: Φi = Φe ,

ϵi

∂Φ i ∂Φ e = ϵe . ∂n ∂n

(9.42)

In the simplest model considered here, the dielectric boundary coincides with the van der Waals surface (see Fig. 9.2). However, the algorithms, which are described here, can be used for more complicated models as well, including the model with the ion-exclusion layer [10, 118]. One of the most important macroparameters for a molecule is its free electrostatic energy. In the linear case, it can be represented as a linear functional of the potential point values: 1 Q m Φ rf (x(m) ). 2 M

W rf =

(9.43)

m=1

Here, Φ rf is the electrostatic potential, which is due to the reaction of electric field on the dielectric boundary and charges outside the molecule. Note that its values have to be computed exactly at the positions of point charges. Remind that the original problem (9.36) is stated in CGS units. At the atomic scale, however, it is natural to measure distances in angstroms (Å) and charges in −1 proton (e) units. In these units, κ is of order 1 Å , while in CGS units, κ is of order 8 −1 10 cm . Therefore, all computations are performed in atomic units. After that, the calculated value of energy in units e2 /Å is multiplied by 332.06364 to get the result measured in kcal/mol.

190 | 9 Macromolecules properties

With the Monte Carlo algorithms, it is possible to separate singularities and to compute the reaction field only. To do this, we use the following representation: Φ i (x) = Φ rf (x) + Φ c (x),

(9.44)

where the singular part can be written down explicitly: Φ c (x) =

M 

Qm . ϵ |x − x(m) | m=1 i

(9.45)

To compute the reaction field potential, we use estimate (9.35), where g = Φ c . Consider now the problem of estimating the dependence of energy on salt concentration in the solution. For linear model and 1:1 salt, this dependence manifests itself through the Debye-Hückel parameter, κ. Let 0 < κ1 < κ2 < . . . < κ nκ be the parameter values, for which the energy should be estimated. Clearly, the survival probability, q(κ, d) (see (9.16)), for the random walk in the outside domain, G e , monotonically decreases when κ increases. Therefore, the minimal value (κ1 ) is used for simulation, while energy estimates for other values of this parameter are calculated using the weight functions, F j (κ): ξ [Φ rf ](x(m) ) = −Φ c (x*1 ) +

N ins 

c * F j (κ) (Φ c (x ins j ) − Φ (x j,ins )),

(9.46)

j=2

where F j (κ) ≡ 1 for κ = κ1 , and at every step in G e , it is multiplied by q(κ i , d k )/q(κ1 , d k ) for all other values of κ. Thus, we have 1 Q m ξ [Φ rf ](x(m) ), 2 M

ξ [W rf ] =

(9.47)

m=1

where salt-dependent estimates for the reaction field potential are given by (9.46). This estimate can be used for computing the electrostatic solvation free energy (∆G solv elec ). To do this, we need to calculate the difference between two results: the first one for the current values of ϵ i , ϵ e and κ and the second (reference) one for ϵ e = 1, κ = 0, i.e., for the molecule in vacuum. In Figure 9.3, we present the results of computations for the particular macromolecule, namely, calmodulin (3cln in the publicly available protein database) in comparison with the FD results. These results fall well within the 95% confidence interval of the Monte Carlo (MC) energy values at all salt concentrations. Moreover, in the limit of zero salt concentration, the salt derivative of ∆G solv elec is equal to zero for both MC- and FD-based PB codes. This property coincides with the theoretically predicted behaviour.

9.5 Computing macromolecule energy

| 191

–7885

FD

solv

ΔGelec (kcal/mole)

–7890

MC –7895

–7900

–7905

–7910

–7915 –6

–5.5

–5

–4.5

–4

–3.5

–3

–2.5

–2

log(salt concentration (M)) Fig. 9.3. Comparison of the salt dependence of the electrostatic solvation free energy (∆G solv ) of elec calmodulin (PDB id: 3cln) obtained with two independent Poisson-Boltzmann (PB) solvers: Monte Carlo estimation (MC) and finite-difference-based software (FD).

It is essential to note that every point of the deterministic curve required a separate PB computation, whereas the Monte Carlo algorithm made it possible to calculate the curve in a single computation.

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